Recent zbMATH articles in MSC 14https://zbmath.org/atom/cc/142023-11-13T18:48:18.785376ZUnknown authorWerkzeugMini-workshop: Subvarieties in projective spaces and their projections. Abstracts from the mini-workshop held November 27 -- December 3, 2022https://zbmath.org/1521.000092023-11-13T18:48:18.785376ZSummary: The major goals of this workshop are to lay paths for a systematic study of geproci (and related, e.g., projecting to almost complete intersections or full intersections) sets of points in projective spaces, study algebraic properties of their ideals (e.g. in the spirit of the Cayley-Bacharach properties), and to identify the most promising new directions for study.Mini-workshop: Quantization of complex symplectic varieties. Abstracts from the mini-workshop held October 2--8, 2022https://zbmath.org/1521.000112023-11-13T18:48:18.785376ZSummary: The mini-workshop featured two main series of lectures: \textit{Functoriality in non-abelian Hodge theory} by Tony Pantev, and \textit{Quantization of the Hitchin system and the analytic Langlands program} by Jörg Teschner. In addition, four senior mathematicians and physicists gave two talks each on their recent mysterious discoveries related to the theme of the workshop. Three junior mathematicians also gave a talk based on their fresh results. All talks by mathematicians and physicists were coordinated to form a common ground of understanding. The smallness of the size of workshop promoted deeper discussions and helped to create friendly and inclusive atmosphere.Algebraic structures in statistical methodology. Abstracts from the workshop held December 4--10, 2022https://zbmath.org/1521.000142023-11-13T18:48:18.785376ZSummary: Algebraic structures arise naturally in a broad variety of statistical problems, and numerous fruitful connections have been made between algebra and discrete mathematics and research on statistical methodology. The workshop took up this theme with a particular focus on algebraic approaches to graphical models, causality, axiomatic systems for independence and non-parametric models.Generic expansion of an abelian variety by a subgrouphttps://zbmath.org/1521.030662023-11-13T18:48:18.785376Z"d'Elbée, Christian"https://zbmath.org/authors/?q=ai:delbee.christianSummary: Let \(A\) be an abelian variety in an algebraically closed field of characteristic 0. We prove that the expansion of \(A\) by a generic divisible subgroup of \(A\) with the same torsion exists provided \(A\) has few algebraic endomorphisms, namely \(\mathrm{End} (A) = \mathbb{Z}\). The resulting theory is \(\mathrm{NSOP}_1\) and not simple. Note that there exist abelian varieties \(A\) with \(\mathrm{End} (A) = \mathbb{Z}\) of any genus.
{{\copyright} 2021 Wiley-VCH GmbH}On \(p\)-adic semi-algebraic continuous selectionshttps://zbmath.org/1521.031032023-11-13T18:48:18.785376Z"Thamrongthanyalak, Athipat"https://zbmath.org/authors/?q=ai:thamrongthanyalak.athipatSummary: Let \(E \subseteq \mathbb{Q}_p^n\) and \(T\) be a set-valued map from \(E\) to \(\mathbb{Q}_p^m\). We prove that if \(T\) is \(p\)-adic semi-algebraic, lower semi-continuous and \(T (x)\) is closed for every \(x \in E\), then \(T\) has a \(p\)-adic semi-algebraic continuous selection. In addition, we include three applications of this result. The first one is related to \textit{C. Fefferman}'s and \textit{J. Kollár}'s [Dev. Math. 28, 233--282 (2013; Zbl 1263.15003)] question on existence of \(p\)-adic semi-algebraic continuous solution of linear equations with polynomial coefficients. The second one is about the existence of \(p\)-adic semi-algebraic continuous extensions of continuous functions. The other application is on the characterization of right invertible \(p\)-adic semi-algebraic continuous functions under the composition.Topological elementary equivalence of regular semi-algebraic sets in three-dimensional spacehttps://zbmath.org/1521.031142023-11-13T18:48:18.785376Z"Geerts, Floris"https://zbmath.org/authors/?q=ai:geerts.floris"Kuijpers, Bart"https://zbmath.org/authors/?q=ai:kuijpers.bart-h-mSummary: We consider semi-algebraic sets and properties of these sets that are expressible by sentences in first-order logic over the reals. We are interested in first-order properties that are invariant under topological transformations of the ambient space. Two semi-algebraic sets are called topologically elementarily equivalent if they cannot be distinguished by such topological first-order sentences. So far, only semi-algebraic sets in one and two-dimensional space have been considered in this context. Our contribution is a natural characterisation of topological elementary equivalence of regular closed semi-algebraic sets in three-dimensional space, extending a known characterisation for the two-dimensional case. Our characterisation is based on the local topological behaviour of semi-algebraic sets and the key observation that topologically elementarily equivalent sets can be transformed into each other by means of geometric transformations, each of them mapping a set to a first-order indistinguishable one.Counting tripods on the torushttps://zbmath.org/1521.050072023-11-13T18:48:18.785376Z"Athreya, Jayadev S."https://zbmath.org/authors/?q=ai:athreya.jayadev-s"Aulicino, David"https://zbmath.org/authors/?q=ai:aulicino.david"Richman, Harry"https://zbmath.org/authors/?q=ai:richman.david-harrySummary: Motivated by the problem of counting finite BPS webs, we count certain immersed metric graphs, tripods, on the flat torus. Classical Euclidean geometry turns this into a lattice point counting problem in \(\mathbb{C}^2\), and we give an asymptotic counting result using lattice point counting techniques.Counting on the variety of modules over the quantum planehttps://zbmath.org/1521.050082023-11-13T18:48:18.785376Z"Huang, Yifeng"https://zbmath.org/authors/?q=ai:huang.yifengSummary: Let \(\zeta\) be a fixed nonzero element in a finite field \(\mathbb{F}_q\) with \(q\) elements. In this article, we count the number of pairs \((A,B)\) of \(n\times n\) matrices over \(\mathbb{F}_q\) satisfying \(AB=\zeta BA\) by giving a generating function. This generalizes a generating function of \textit{W. Feit} and \textit{N. J. Fine} [Duke Math. J. 27, 91--94 (1960; Zbl 0097.00702)] that counts pairs of commuting matrices. Our result can be also viewed as the point count of the variety of modules over the quantum plane \(XY=\zeta YX\), whose geometry was described by \textit{X. Chen} and \textit{M. Lu} [J. Algebra 580, 158--198 (2021; Zbl 1507.16015)].Tableau formulas for skew Schubert polynomialshttps://zbmath.org/1521.052132023-11-13T18:48:18.785376Z"Tamvakis, Harry"https://zbmath.org/authors/?q=ai:tamvakis.harrySummary: The skew Schubert polynomials are those that are indexed by skew elements of the Weyl group, in the sense of \textit{H. Tamvakis} [J. Reine Angew. Math. 652, 207--244 (2011; Zbl 1276.14082)]. We obtain tableau formulas for the double versions of these polynomials in all four classical Lie types, where the tableaux used are fillings of the associated skew Young diagram. These are the first such theorems for symplectic and orthogonal Schubert polynomials, even in the single case. We also deduce tableau formulas for double Schur, double theta, and double eta polynomials, in their specializations as double Grassmannian Schubert polynomials. The latter results generalize the tableau formulas for symmetric (and single) Schubert polynomials due to Littlewood (in type A) and the author (in types B, C, and D).
{{\copyright} 2023 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}A curious identity that implies Faber's conjecturehttps://zbmath.org/1521.052142023-11-13T18:48:18.785376Z"Garcia-Failde, Elba"https://zbmath.org/authors/?q=ai:garcia-failde.elba"Zagier, Don"https://zbmath.org/authors/?q=ai:zagier.don-bernardSummary: We prove that a curious generating series identity implies \textit{C. Faber}'s intersection number conjecture [in: Moduli of curves and abelian varieties. The Dutch intercity seminar on moduli. Braunschweig: Vieweg. 109--129 (1999; Zbl 0978.14029)] (by showing that it implies a combinatorial identity already given in \textit{E. Garcia-Failde} et al. [SIGMA, Symmetry Integrability Geom. Methods Appl. 15, Paper 080, 27 p. (2019; Zbl 1498.14073)] and give a new proof of Faber's conjecture by directly proving this identity.
{{\copyright} 2022 The Authors. \textit{Bulletin of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.}Supersingular \(p\)-adic \(L\)-functions, Maass-Shimura operators and Waldspurger formulashttps://zbmath.org/1521.110012023-11-13T18:48:18.785376Z"Kriz, Daniel J."https://zbmath.org/authors/?q=ai:kriz.daniel-jThis book presents a unification of classical and modern results in the number theory. It is dedicated to the development of a completely new theory of \(p\)-adic modular forms on modular curves, thus extending the classical theory, primarily due to Katz, to the supersingular locus.
In general, theory of \(p\)-adic \(L\)-functions plays an important role as the refinement of the study of special values of \(L\)-functions and their arithmetic. As such, it has been widely used in the number theory for decades, providing a transfer of the arithmetic information along families of the Galois representations through the interpolation. Some of the most important arithmetic applications of \(p\)-adic \(L\)-functions appear in the so-called Rankin-Selberg settings. The very first examples of the Rankin-Selberg type \(p\)-adic \(L\)-functions appear in the work of Katz in the \('70\)s, which have been generalized in several directions over the last \(20\) years, followed by the establishing of the \(p\)-adic Waldspurger formula in the corresponding settings.
However, in all mentioned results constructions of the Rankin-Selberg type \(p\)-adic \(L\)-functions rely on the assumption that \(p\) is split in the observed field \(K\), which implies that the associated automorphic representations in the critical locus are ordinary. Thus, several groundbreaking results in number theory are at the moment restricted to the ordinary settings.
The aim of the book under the review is to introduce the \(p\)-adic \(L\)-functions of the Rankin-Selberg type which allow \(p\) to be inert of ramified in \(K\), i.e., do not have to satisfy the ordinariness assumption. The author defines generalized \(p\)-adic modular forms as sections of relative period sheaves transforming under the Galois group of the modular curve by weight characters. He develops a new and more general theory of \(p\)-adic modular forms, realizing them as sections of de Rham period sheaves.
The classical \(\theta\)-operator is generalized to the notion of \(p\)-adic Maass-Shimura operator, which raises the weights of generalized \(p\)-adic modular forms. A detailed analysis of this operator is presented and it is explained how it gives rise to \(p\)-adic \(L\)-functions of supersingular Rankin-Selberg type. The \(p\)-adic Waldspurger formula for the supersingular \(p\)-adic \(L\)-functions is also obtained.
The book contains a reasonable amount of the background material and can be studied by both experts and novices in the field. Presented results, beside being interesting by themselves, might also be of a great importance for future development of some aspects of the number theory.
Chapter 1 provides an introduction, while in the Chapter 2 some background on the used theories, such as category theory and Huber's theory, is given. Key properties of the geometry of modular curves are listed in the Chapter 3. In Chapter 4 the author introduces the notion of the fundamental de Rham periods, which are essential for the provided construction. A construction of the \(p\)-adic Maass-Shimura operator and of generalized \(p\)-adic modular forms is given in Chapter 5, while some of the important properties of the introduced operator are studied in Chapter 6. Bounding periods at supersingular CM points is provided in the Chapter 7, which enables a construction of the supersingular Rankin-Selberg type \(p\)-adic \(L\)-functions, given in Chapter 8. In the final chapter the \(p\)-adic Waldspurger formula is established.
Reviewer: Ivan Matić (Osijek)Hecke operators and systems of eigenvalues on Siegel cusp formshttps://zbmath.org/1521.110042023-11-13T18:48:18.785376Z"Hatada, Kazuyuki"https://zbmath.org/authors/?q=ai:hatada.kazuyukiThis texts conducts a rigorous study of Hecke operators with a view to structuring the associated eigenvalues, in the context of holomorphic Siegel modular forms of arbitrary degree. By expressing the Hecke operators associated with the principal congruence subgroup of $\mathrm{Sp}(g,\mathbb{Z})$ in terms of endomorphisms of certain rigid analytic varieties in terms of precisely defined coset operations, the authors manage to give an estimate for any eigenvalue of the Hecke operator under consideration, in the form of a $p$-expansion. This construction allows them to show the existence of a Siegel cusp eigenform in every non zero space of holomorphic cusp forms of weight greater than the genus $g$. This explicit construction enables the imposition of an upper bound on the absolute value of the Fourier coefficients of a given Siegel cusp eigenform, and consequently the authors discover that there are infinitely many holomorphic Siegel cusp eigenforms of degree $g$ that satisfy an upper bound rule set by the generalised Ramanujan or Ramanujan-Petersson conjecture. This has exciting consequences and implications especially in light of recent number Theoretic results on Rademacher expansions [\textit{G. L. Cardoso}, ``Rademacher expansion of a Siegel modular form for $N=4$ counting'', Preprint, \url{arXiv:2112.10023}] for the inverse of Siegel forms with very clear formulas expressing the Siegel Fourier coefficients in terms of an infinite convergent sum of Bessel functions.
Given that these coefficients, in certain case encode the statistical degeneracy of BPS black holes in string theory, proving their positivity becomes vital. In light of the fact that these Siegels forms, notably the Igusa cusp form of weight 10, have both an arithmetic and multiplicative lift construction in terms of Hecke operators, the systematisation of eigenvalues in the paper under review as well as the result pertaining to upper bound constraints obeying the generalised Ramanujan conjecture will be critical in determining the positivity of Fourier coefficients of the inverse Siegel forms of interest.
Reviewer: Suresh Nampuri (Lisboa)Potential automorphy over CM fieldshttps://zbmath.org/1521.110342023-11-13T18:48:18.785376Z"Allen, Patrick B."https://zbmath.org/authors/?q=ai:allen.patrick-b"Calegari, Frank"https://zbmath.org/authors/?q=ai:calegari.frank"Caraiani, Ana"https://zbmath.org/authors/?q=ai:caraiani.ana"Gee, Toby"https://zbmath.org/authors/?q=ai:gee.toby"Helm, David"https://zbmath.org/authors/?q=ai:helm.david"Le Hung, Bao"https://zbmath.org/authors/?q=ai:le-hung.bao-v"Newton, James"https://zbmath.org/authors/?q=ai:newton.james"Scholze, Peter"https://zbmath.org/authors/?q=ai:scholze.peter"Taylor, Richard"https://zbmath.org/authors/?q=ai:taylor.richard-l"Thorne, Jack A."https://zbmath.org/authors/?q=ai:thorne.jack-aIn this so-called ``ten author paper'', the authors prove that the Sato-Tate conjecture holds for any elliptic curve over a CM field.
Let us give some background. \textit{F. Calegari} and \textit{D. Geraghty} [Invent. Math. 211, No. 1, 297--433 (2018; Zbl 1476.11078)] had an insight for how to extend the Taylor-Wiles method for modularity lifting to a very general setting. This depends on two conjectures about torsion classes in the cohomology of locally symmetric spaces. The first conjecture was that the Galois representations constructed by Scholze satisfy a strong form of local-global compatibility at all primes. The second was a vanishing conjecture for the mod-\(p\) cohomology of arithmetic groups localized at non-Eisenstein primes that mirrored the corresponding vanishing theorems for cohomology corresponding to tempered automorphic representations in characteristic zero.
In the present paper, the authors implement the Calegari-Geraghty method unconditionally. They prove the first unconditional modularity lifting theorems for \(n\)-dimensional regular Galois representations without any self-duality conditions. In fact, they prove many cases of the first of these conjectures. Their arguments crucially exploit work of Caraiani and Scholze on the cohomology of non-compact Shimura varieties. On the other hand, they do not resolve the second conjecture concerning the vanishing of mod-\(p\) cohomology. Instead, they sidestep this difficulty by a new technical innovation: a derived version of Ihara avoidance. In particular, they prove the local-global compatibility for \(l \neq p\) or \(l=p\) in both the ordinary and Fontaine-Laffaille case. They obtain quite general modularity lifting theorems in both the ordinary and Fontaine-Laffaille case for general \(n\)-dimensional representations over CM fields. As an application, they show that if \(E\) is an elliptic curve over a CM number field \(F\), then \(E\) and all the symmetric powers of \(E\) are potentially modular. Consequently, the Sato-Tate conjecture holds for \(E\). As an application of a different sort, they also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for \(\mathrm{GL}_2(\mathbb{A}_F)\).
Reviewer: Lei Yang (Beijing)Ramification in division fields and sporadic points on modular curveshttps://zbmath.org/1521.110392023-11-13T18:48:18.785376Z"Smith, Hanson"https://zbmath.org/authors/?q=ai:smith.hansonLet \(E\) be an elliptic curve defined over a number field \(K\). Let \(\mathfrak p\) be a prime ideal of \(K\) lying over a rational prime \(p\). Let \(\nu\) be the valuation of the \(p^n\)-division field \(K_n\) of \(E\) normalized as \(\nu(p)=1\). Let \(E\) have supersingular reduction at \(\mathfrak p\). The author classifies the valuations of the \(p^n\)-torsion points of \(E\) and gives a lower bound of the ramification index of \(\mathfrak p\) over \(p\) and that of \(p\) in \(K_n\) respectively. Since \(p^n\)-torsion points are in the kernel of the reduction map modulo \(\mathfrak p\), the valuation can be computed by using the formal group \(\hat{E}\) of \(E\) at \(\mathfrak p\) and the \(p^n\)-th division polynomial. If there exist \(R,Q\ne 0\) in \(\hat{E}[p]\) such that \(\nu(R)>\nu (Q)\), then the subgroup \(\langle R\rangle\) is called the (level-\(1\)) canonical subgroup at \(\mathfrak p\). He defines the level-\(n\) canonical subgroup as the generalization. Let \(\mu\) be the valuation of the coefficient of \(x^{(p^2-p)/2}\) in the \(p\)-th division polynomial of \(E\) if \(p\) is odd and \(\mu= \nu(b_2)/2\) (\(b_2\) is the standard invariant of \(E\)) if \(p=2\). The valuations are classified by \(\mu\) and the existence of the level \(n\)-canonical group in Theorem 4.6. For example, if \(\mu\ge\frac{p}{p+1}\), then there is no canonical subgroup and \(\nu(\hat{P})=1/(p^{2n}-p^{2n-2})\) for \(\hat{P}\) of exact order \(p^n\). Taking \(\nu(p)=1\), the denominator of \(\nu(\hat{P})\) implies the ramification index. He shows that \(E\) does not have a canonical subgroup if \(\mathfrak p\) unramified over \(p\). Let \(N\in\mathbb Z_{>0}\). Let \(E/\mathbb Q\) have supersingular reduction at all prime divisors \(p_i\) of \(N\). If \(E\) has a point of exact order \(N\) over a number field \(L\). Then any \(p_i\) in \(L\) has ramification index divisible by \(p_i^{2n_i}-p_i^{2n_i-2} ~(p^{n_i}|| N)\) and \(N^2\prod (1-p_i^{-2})\) divides \([L:\mathbb Q]\). This gives a lower bound of the degree of \(L\). As an application, he considers sporadic points on modular curves. Let \(k\) be the smallest positive integer such that there are infinitely many points of degree \(k\) on \(X_1(N)\). The number \(k\le \frac{11N^2}{840}\). A point on \(X_1(N)\) of degree \(<k\) is called a sporadic point. The degree of \(x\in X_1(N)\) corresponding to the pair \((E,P)\) is \([\mathbb Q(j(E),h(P)):\mathbb Q]\), where \(j(E)\) is the \(j\)-invariant of \(E\) and \(h\) is a Weber function for \(E\). By comparing a lower bound of degree of \(x\) with an upper bound of \(k\), he shows that \(j(E)\) dose not correspond to a sporadic point on \(X_1(p^n)\) for any \(n>0\), if \(E\) does not have a canonical subgroup at \(\mathfrak p\). Further if \(N=\prod_{i=1}^kp_i^{n_i}>12\) is not divisible by \(6\) and \(E/\mathbb Q\) has supersingular reduction at each \(p_i\), then \(j(E)\) does not corresponds to a sporadic point.
Reviewer: Noburo Ishii (Kyōto)Anderson \(t\)-motives and abelian varieties with MIQF: results coming from an analogyhttps://zbmath.org/1521.110412023-11-13T18:48:18.785376Z"Grishkov, A."https://zbmath.org/authors/?q=ai:grishkov.alexander-n"Logachev, D."https://zbmath.org/authors/?q=ai:logachev.dmitryThe aim of this paper is the study of some analogies between Anderson \(t\)-motives (function field case) and abelian varieties (number field case). Usually, the analogies are from abelian varieties to Anderson \(t\)-motives. In this article the direction of the analogies is from Anderson \(t\)-motives to abelian varieties. An abelian variety \(A\) over \({\mathbb C}\) with multiplication by an imaginary quadratic field \(K={\mathbb Q}(\sqrt{-\Delta})\), of dimension \(r\) and of signature \((n,r-n)\) is called an abelian variety with MIQF \(K\), while MIQF-PP means an abelian variety with MIQF having \(T\) such that \(T=-\sqrt{-\Delta}\ J_{n,r-n}\), where \(J_{n.r-n}=\Big(\begin{smallmatrix} I_n&0\\ 0&-I_{r-n}\end{smallmatrix}\Big)\).
Let \(A=V/D\), where \(V={\mathbb C}^r\) and \(D={\mathbb Z}^{2r}\), be an abelian variety with MIQF, and \({\mathfrak D}:=D\otimes_{\mathbb Z}{\mathbb Q}\). Fix an inclusion \(\iota\colon K \hookrightarrow{\mathrm{End}}(A) \otimes_{\mathbb Z}{\mathbb Q}\) defining multiplication and an Hermitian polarization form \(H:=B+iE\) of \(A\) on \(V\), where \(B\) and \(E\) are, respectively, its real and imaginary parts.
Let \(L\) be a \(K\)-vector space of dimension \(r\) and \(H_L\) a \(K\)-valued Hermitian form on \(L\) of signature \((n,r-n)\) such that there exists a basis of \(L\) over \(K\) where the matrix of \(H_L\) is \(J_{n,r-n}\) and \(\alpha\colon L\otimes_K{\mathbb C}\to {\mathbb C}^n\) is a \({\mathbb C}\)-linear map such that \(\alpha\) is surjective and the restriction of \(-H_{L,{\mathbb C}}\) to \(\ker \alpha\) is a positive definite form.
The main result of this paper is that there is a 1-1 correspondence between \(V,{\mathfrak D},K,\iota, H\) and the triplet \((L,H_L,\alpha)\).
Finally, the triplet \((L,H_L,\alpha)\) defines an exact sequence of \({\mathbb C}\)-vector spaces
\[
0\to \ker \alpha\stackrel{\iota}{\hookrightarrow}L\otimes_K {\mathbb C}\stackrel{\alpha}{\to}{\mathbb C}^n\to 0.
\]
The \(k\)-th exterior powers \(\lambda^k(L),(-1)^{k-1}\lambda^k(H_L), \alpha_k\) satisfy the conditions of the main result and therefore define an abelian variety with MIQF-PP (up to isogeny) \(\lambda^k (A)\), the \(k\)-th exterior power of \(A\). Its signature is \(\binom{r-1} {k-1},\binom{r-1}k\).
Reviewer: Gabriel D. Villa Salvador (Ciudad de México)Real quadratic Borcherds productshttps://zbmath.org/1521.110422023-11-13T18:48:18.785376Z"Darmon, Henri"https://zbmath.org/authors/?q=ai:darmon.henri"Vonk, Jan"https://zbmath.org/authors/?q=ai:vonk.janClassically, abelian extensions of quadratic imaginary number fields are generated by values of modular and elliptic functions at special points on the upper half plane. In analogy to this situation, in a previous paper [\textit{H. Darmon} and \textit{J. Vonk}, Duke Math. J. 170, No. 1, 23--93 (2021; Zbl 1486.11137)] the authors constructed \textit{rigid meromorphic cocycles} whose values at special points on Drinfeld's \(p\)-adic upper half plane (for a fixed prime \(p\)) generate abelian extensions of real quadratic number fields.
In the current paper, this analogy between imaginary and real quadratic number fields is developed further. Certain modular forms of weight \(\frac{1}{2}\) are lifted to the group of rigid meromophic cocycles with rational real multiplication divisor, in a manner analogous to Borcherd's lifts in the complex multiplication case.
Reviewer: Salman Abdulali (Greenville)Points of bounded height on curves and the dimension growth conjecture over \(\mathbb{F}_q[t]\)https://zbmath.org/1521.110442023-11-13T18:48:18.785376Z"Vermeulen, Floris"https://zbmath.org/authors/?q=ai:vermeulen.florisSummary: In this article, we prove several new uniform upper bounds on the number of points of bounded height on varieties over \(\mathbb{F}_q[t]\). For projective curves, we prove the analogue of Walsh' result with polynomial dependence on \(q\) and the degree \(d\) of the curve. For affine curves, this yields an improvement to bounds by \textit{A. Sedunova} [Acta Arith. 181, No. 4, 321--331 (2017; Zbl 1422.11203)], and \textit{R. Cluckers} et al. [Algebra Number Theory 14, No. 6, 1423--1456 (2020; Zbl 1442.11098)]. In higher dimensions, we prove a version of dimension growth for hypersurfaces of degree \(d\geqslant 64\), building on work by \textit{W. Castryck} et al. [Algebra Number Theory 14, No. 8, 2261--2294 (2020; Zbl 1480.11083)] in characteristic zero. These bounds depend polynomially on \(q\) and \(d\), and it is this dependence which simplifies the treatment of the dimension growth conjecture.On the Northcott property of zeta functions over function fieldshttps://zbmath.org/1521.110452023-11-13T18:48:18.785376Z"Généreux, Xavier"https://zbmath.org/authors/?q=ai:genereux.xavier"Lalín, Matilde"https://zbmath.org/authors/?q=ai:lalin.matilde-n"Li, Wanlin"https://zbmath.org/authors/?q=ai:li.wanlinThe Northcott property [\textit{D. G. Northcott}, Proc. Camb. Philos. Soc. 45, 502--509, 510--518 (1949; Zbl 0035.30701)] implies that a set of algebraic numbers with bounded height an bounded degree is finite. \textit{F. Pazaki} and \textit{R. Pengo} [``On the Northcott property for special values of \(L\)-functions'', Preprint. \url{arXiv:2012.00542}] defined a Northcott property for special values of zeta functions of number fields and certain motivic \(L\)-functions.
In this paper, the authors explore the Northcott property for global function fields. Consider the set isomorphism classes of function fields \(K\) with constant field \({\mathbb F}_q\). Let \(\zeta_K^*(s):=\lim_{t\to s}\frac{\zeta_K(t)}{(t-s)^{\mathrm{ord}_s( \zeta_K(t))}}\) be the first nonzero coefficient for the Taylor series for the zeta function \(\zeta_K(s)\) around \(s\). Set \(S_{q,s,B}=\{[K]\mid |\zeta_K^*(s)|\leq B\}\), where \([K]\) denotes the isomorphism class of \(K\). The Northcott property of \({\mathbb F}_q\) at \(s\) is equivalent to having \(S_{q,s,B}\) finite for all positive real numbers \(B\).
The authors determine the values for which the Northcott property holds outside the critical strip. Then, they study case by case for some values inside the critical strip, notably \(\Re(s)<\frac 12- \frac{\log 2}{\log q}\) and for \(s\) real such that \(1/2\leq s \leq 1\) and obtain a partial result for complex \(s\) with \(1/2 < \Re(s)\leq 1\). This is the content of the main result, Theorem 1.3.
Sections 3 and 4 treat the left and right sides of the critical strip respectively, and Section 5 considers the critical strip.
Reviewer: Gabriel D. Villa Salvador (Ciudad de México)Schinzel hypothesis on average and rational pointshttps://zbmath.org/1521.110572023-11-13T18:48:18.785376Z"Skorobogatov, Alexei N."https://zbmath.org/authors/?q=ai:skorobogatov.alexei-nikolaievitch"Sofos, Efthymios"https://zbmath.org/authors/?q=ai:sofos.efthymiosCall \(P(t) \in \mathbb{Z}[t]\) a Bouniakowsky polynomial if the leading coefficient of \(P(t)\) is positive and for every prime \(p\) the reduction of \(P(t)\) modulo \(p\) is not a multiple of \(t^p-t\). Schinzel's Hypothesis (H) asserts that \(P(n)\) represents infinitely many primes as \(n\) runs over the integers if \(P(t)\) is an irreducible Bouniakowsky polynomial. The Bateman-Horn conjecture is a quantitative form of this hypothesis, formulated, more in general, for \(n\)-tuples of polynomials. For applications it is often enough to know that a Bouniakowsky polynomial represents at least one prime. The authors prove that this is true for almost all Bouniakowsky polynomials of any fixed degree \(d\). More precisely, the proportion of Bouniakowsky polynomials of degree \(d\) with this property tends to 100\% as the height tends to infinity. The authors also derive a generalization of this result for \(n\)-tuples of polynomials under certain congruence restrictions. As applications, they present several results on the existence of \(\mathbb{Q}\)-rational points on varieties in families. In particular, they deduce that a positive proportion of diagonal conic bundles over \(\mathbb{Q}\) with any given number of degenerate fibres have a rational point. To prove their main result on Bouniakowsky polynomials, the authors establish a vast generalisation of the Barban-Davenport-Halberstam theorem on primes in arithmetic progressions. The relevant result estimates an average of the error term in the Bateman-Horn conjecture. Its proof is a generalization of Montgomery's proof of the Barban-Davenport-Halberstam theorem.
Reviewer: Stephan Baier (Hāoṛā)Corrigendum to: ``Kummer-faithful fields which are not sub-\(p\)-adic''https://zbmath.org/1521.110692023-11-13T18:48:18.785376Z"Ohtani, Sachiko"https://zbmath.org/authors/?q=ai:ohtani.sachikoSummary: We correct errors in our previous paper [ibid. 8, No. 1, Paper No. 15, 7 p. (2022; Zbl 1518.11076)]. In more detail, we first correct the proof of [loc. cit., Theorem 2]. Second, we modify the statements and the proofs of [loc. cit., Theorem 3, Proposition 1]. Third, we fix the proof of [loc. cit., Proposition 2] by using the modified version of [loc. cit., Theorem 3] and correct the statement of [loc. cit., Corollary 1]. Fourth, we correct the proof of [loc. cit., Proposition 3]. Finally, we modify the construction of [loc. cit., Example 2].The Massey vanishing conjecture for number fieldshttps://zbmath.org/1521.110702023-11-13T18:48:18.785376Z"Harpaz, Yonatan"https://zbmath.org/authors/?q=ai:harpaz.yonatan"Wittenberg, Olivier"https://zbmath.org/authors/?q=ai:wittenberg.olivierHigher Massey products are invariants of a differential graded ring that do not factor through its cohomology. As such, they detect differences that are usually not seen by classical cohomological methods. The original example studied by Massey was that of the complement in \(\mathbb{R}^3\) of the Borromean ring configuration [\textit{W. S. Massey}, in: Conf. algebr. Topol., Univ. Ill. Chicago Circle 1968, 174--205 (1969; Zbl 0212.55904); \textit{W. S. Massey}, J. Knot Theory Ramifications 7, No. 3, 393--414 (1998; Zbl 0911.57009)]. Here, cohomology with coefficients in \(\mathbb{Z}\) is not able to detect the difference between this space and the complement of three completely unlinked rings, but higher Massey products in the corresponding differential complex do capture the essence of this nontrivial configuration.
These constructions were later studied in a more arithmetic context, considering étale cohomology of open subschemes of spectra of Dedekind rings, and finally taken to the ``simpler'' context of Galois cohomology with coefficients in \(\mathbb{F}_p\), where \(p\) is a prime number. Here, a higher Massey product takes classes \(\alpha_1,\ldots,\alpha_n\in H^1(k,\mathbb{F}_p)\), with \(n\geq 3\), and gives back a subset of \(H^2(k,\mathbb{F}_p)\), which could eventually be empty. It is in this context that Mináč and Tân conjectured that higher Massey products over an arbitrary field \(k\) contain the trivial class \(0\in H^2(k,\mathbb{F}_p)\) as soon as they are non-empty (in which case one says that the product \emph{vanishes}). This is now known as the \emph{Massey vanishing conjecture}.
In this article, the authors prove that the Massey vanishing conjecture holds for number fields (cf.~Theorem 1.3). This was known to be true when \(n=3\) (and for an arbitrary field \(k\), cf.~for instance [\textit{J. Mináč} and \textit{N. D. Tân}, J. Lond. Math. Soc., II. Ser. 94, No. 3, 909--932 (2016; Zbl 1378.12002)]) and for \(n=4\) and \(p=2\) [\textit{P. Guillot} et al., Compos. Math. 154, No. 9, 1921--1959 (2018; Zbl 1455.12005)]; otherwise the question was completely open. This is thus a great breakthrough that hinges on:
\begin{itemize}
\item a translation of this conjecture in terms of embedding problems, due to \textit{W. G. Dwyer} [J. Pure Appl. Algebra 6, 177--190 (1975; Zbl 0338.20057)];
\item a translation of embedding problems in terms of homogeneous spaces of \(\mathrm{SL}_n\), due to \textit{A. Pál} and \textit{T. M. Schlank} [Int. J. Number Theory 18, No. 7, 1535--1565 (2022; Zbl 1500.12005)];
\item recent developments by the authors of this paper in the study of the Brauer-Manin obstruction to the local-global principle for homogeneous spaces with finite supersolvable stabilizers [\textit{Y. Harpaz} and \textit{O. Wittenberg}, J. Am. Math. Soc. 33, No. 3, 775--805 (2020; Zbl 1469.14053)]; and
\item known results on the Massey vanishing conjecture over local fields [\textit{J. Mináč} and \textit{N. D. Tân}, J. Eur. Math. Soc. (JEMS) 19, No. 1, 255--284 (2017; Zbl 1372.12004)] and refinements of these by the authors of this paper (cf.~in particular Proposition 5.3).
\end{itemize}
More precisely, given classes \(\alpha_1,\ldots,\alpha_n\in H^1(k,\mathbb{F}_p)\), the first two tools provide a homogeneous space \(V\) of \(\mathrm{SL}_n\) with finite geometric stabilizers that is a \emph{splitting variety} for the vanishing of the corresponding Massey product (cf.~Proposition 2.5). This means that \(V(L)\neq\emptyset\) if and only if the Massey product vanishes when restricted to \(H^2(L,\mathbb{F}_p)\). Since the stabilizers of this homogeneous space turn out to be \emph{supersolvable}, this enables the authors to apply their machinery on the Brauer-Manin obstruction and thus concentrate on whether this homogeneous space has local points (which it does, as a nice consequence of local duality) and whether these local points define adelic points that are orthogonal to the \emph{unramified Brauer group} \(\mathrm{Br}_{\mathrm{nr}}(V)\). This second part requires delicate computations of the group \(\mathrm{Br}_{\mathrm{nr}}(V)\), which rely on the explicit description of the stabilizers of this homogeneous space and on previous results by \textit{F. A. Bogomolov} (for the ``geometric'' part of the Brauer group, [Math. USSR, Izv. 30, No. 3, 455--485 (1988; Zbl 0679.14025); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 3, 485--516 (1987)]) and by Harari, Demarche and Lucchini Arteche (for the ``algebraic'' part, cf. [\textit{D. Harari}, Bull. Soc. Math. Fr. 135, No. 4, 549--564 (2007; Zbl 1207.11048); \textit{C. Demarche}, Math. Ann. 346, No. 4, 949--968 (2010; Zbl 1297.14022); \textit{G. Lucchini Arteche}, J. Algebra 411, 129--181 (2014; Zbl 1368.14033)]).
The last two sections of the article (6 and 7) are devoted to further computations of Brauer groups of these splitting varieties in particular cases, as well as generalizations of the Massey vanishing conjecture to cohomology with other coefficients (i.e.,~different from \(\mathbb{F}_p\)) for which the same methods apply.
Reviewer: Giancarlo Lucchini Arteche (Santiago)Companion varieties for Hesse, Hesse union dual Hesse arrangementshttps://zbmath.org/1521.130052023-11-13T18:48:18.785376Z"De Poi, Pietro"https://zbmath.org/authors/?q=ai:de-poi.pietro"Ilardi, Giovanna"https://zbmath.org/authors/?q=ai:ilardi.giovannaIn the paper under review, the authors study the so-called companion varieties for some line arrangements over the complex numbers. The starting point is the notion of unexpected hypersurfaces.
Definition. We say that a reduced set of points \(Z \subset \mathbb{P}^{N}\) admits an unexpected hypersurface of degree \(d\) if there exists a sequence of non-negative integers \(m_{1}, \dots m_{s}\) such that for all general points \(P_{1}, \dots P_{s}\) the zero dimensional subscheme \(P=m_{1}P_{1} + \dots + m_{s}P_{s}\) fails to impose independent conditions on forms of degree \(d\) vanishing along \(Z\) and the set of such forms is non-empty.
Assume now that there is a set of points \(Z \subset \mathbb{P}^{N}\) which admits a unique unexpected hypersurface \(H_{Z,P}\) of degree \(d\) and multiplicity \(m\) at a general point \(P=(a_{0}: \dots : a_{N}) \in \mathbb{P}^{N}\). Let \[F_{Z}((x_{0}:\, \dots \,: x_{N}),(a_{0}:\, \dots \,: a_{N})) = 0\] be a homogeneous polynomial equation of \(H_{Z,P}\). Let \(g_{0}, \dots , g_{M}\) be a basis of the vector space \([I(Z)]_{d}\) of homogeneous polynomials of degree \(d\) vanishing at all points of \(Z\). Under some reasonable conditions the unexpected hypersurface \(H_{Z,P}\) comes from a bi-homogeneous polynomial \(F_{Z}((x_{0}: \dots : x_{N}),(a_{0}: \dots : a_{N}))\) of bi-degree \((m,d)\). Indeed, \(F_{Z}\) can be written in a unique way as a combination \[(\star): \quad F_{Z} = h_{0}(a_{0} : \dots : a_{N})g_{0}(x_{0} : \dots : x_{N}) + \cdots + h_{M}(a_{0} : \dots : a_{N})g_{M}(x_{0} : \dots : x_{N}),\] where \(g_{0}(x_{1}: \dots : x_{N})\), \dots , \(g_{M}(x_{0} : \dots x_{N})\) are homogeneous polynomials of degree \(d\) and \(h_{0}(a_{0} : \dots : a_{N})\), \dots , \(h_{M}(a_{0}: \dots : a_{N})\) are homogeneous polynomials of degree \(m\). Therefore, there are two rational maps naturally associated with \((\star)\), namely \[\phi : \mathbb{P}^{N} \ni (x_{0} : \dots : x_{N}) \mapsto(g_{0}(x_{0}: \dots : x_{N}): \dots : g_{M}(x_{0}: \dots : x_{N})) \in \mathbb{P}^{N},\] \[\psi : \mathbb{P}^{N} \ni (a_{0} : \dots : a_{N}) \mapsto(h_{0}(a_{0}: \dots : a_{N}): \dots : h_{M}(a_{0}: \dots : a_{N})) \in \mathbb{P}^{N}.\] The images of these maps are the companion varieties. The main result of the paper under review can be formulated as follows.
Main Theorem. The image \(S\) of \(\phi\) is a smooth arithmetically Cohen-Macaulay rational surface in the case of the Hesse and the merger of the Hesse and the dual Hesse arrangements.
1) In the case of the Hesse arrangement, the surface \(S\) is of degree \(13\). More precisely, it is the plane blow up in the \(12\) points of \(Z(\mathrm{Hesse})\) (see system (2) therein for details) embedded into \(\mathbb{P}^{8}\) with the complete linear system of the quintics through \(Z(\mathrm{Hesse})\). Its ideal \(I(S)\) is generated by \(15\) quadrics.
2) In the case of the merger of the Hesse and the dual Hesse, the surface \(S\) is of degree \(43\). More precisely, it is the plane blown-up in the \(21\) points of \(Z(\mathrm{Hesse} \cup \mathrm{dualHesse})\) (see system (6) therein for details), embedded into \(\mathbb{P}^{23}\) with the complete linear system of the octics through \(Z(\mathrm{Hesse} \cup\mathrm{dualHesse})\). Its ideal \(I(S)\) is generated by \(210\) quadrics.
Reviewer: Piotr Pokora (Kraków)When does a perturbation of the equations preserve the normal cone?https://zbmath.org/1521.130082023-11-13T18:48:18.785376Z"Quy, Pham Hung"https://zbmath.org/authors/?q=ai:pham-hung-quy."Trung, Ngo Viet"https://zbmath.org/authors/?q=ai:ngo-viet-trung.Let \(I = (f_1,\ldots,f_r)\) denote an ideal of a local Noetherian ring \((R,\mathfrak{m})\). An ideal \(I' = (f_1',\ldots,f_m')\) of \(R\) is called a small pertubation of \(I\) whenever \(f_i \cong f_i' \mod \mathfrak{m}^N\) for \(i=1,\ldots,m\) and \(N \gg 0\). It is of some interest to know which properties of \(R/I\) are preserved under small pertubations. For instances when defining ideal of the singularity of an analytic space is replaced by their \(N\)-jets for some \(N \gg 0\). The paper deals with the following problem: Let \(J \subset R\) denote an arbitrary ideal. For which ideal \(I\) does there exists a number \(N\) such that \(f_i \cong f_i' \mod J^N\) for \(i=1,\ldots,m\) such that \(\operatorname{gr}_J(R/I) \cong \operatorname{gr}_J(R/I')\)? (Note that \(\operatorname{Spec} (\operatorname{gr}_J(R/I) \) is the normal cone of the blow-up of \(R/I\) along \(J\).) -- In their main result the authors provide an affirmative answer when \(f_1,\ldots,f_m\) is a \(J\)-filter regular sequence, i.e. \((f_1,\ldots,f_{i-1}) :_R f_i /(f_1,\ldots,f_{i-1})\) is of \(J\)-torsion for \(i = 1,\ldots,m\). Moreover, they prove a converse to the result by assuming that \(\operatorname{gr}_J(R/(f_1,\ldots,f_i)) \cong \operatorname{gr}_J(R/(f_1',\ldots,f_i'))\) for \(i = 1,\ldots,m\). Finally they generalize some of their results to Noetherian filtrations.
Reviewer: Peter Schenzel (Halle)The algebraic cohomotopy group and its propertieshttps://zbmath.org/1521.130142023-11-13T18:48:18.785376Z"Sridharan, Raja"https://zbmath.org/authors/?q=ai:sridharan.raja"Upadhyay, Sumit Kumar"https://zbmath.org/authors/?q=ai:kumar-upadhyay.sumit"Yadav, Sunil Kumar"https://zbmath.org/authors/?q=ai:yadav.sunil-kumar|kumar-yadav.sunilFor a commutative ring \(A\) with unity, a row \(\mathbf a=(a_1,\dots,a_n)\) of ring elements is called unimodular of length \(n\), if there is another row \(\mathbf b=(b_1,\dots,b_n)\) of elements in \(A\) such that their dot product \(\mathbf a\cdot \mathbf b=1\); the set of unimodular rows of \(A\) is denoted by \(\mathrm{Um}_n(A)\). The paper considers the case \(n=2\).
An equivalence relation \(\sim\) on \(\mathrm{Um}_2(A)\) is defined by either of the following two equivalent conditions: (1) There exists \((f_1(T), f_2(T))\in\mathrm{Um}_2(A[T])\) such that \((f_1(0),f_2(0))=(a,b)\) and \((f_1(1),f_2(1))=(c,d)\) (here \(A[T]\) is the polynomial ring over \(A\) with variable \(T\)); (2) There is a matrix \(\alpha\in\mathrm{SL}_2(A)\) for which there exists a matrix \(\beta(T)\in\mathrm{SL}_2(A[T])\) with \(\beta(0)=I_2\) and \(\beta(1)=\alpha\), such that \(\alpha(a, b)^t=(c, d)^t\).
The equivalence classes \(\mathrm{Um}_2(A)/_\sim = \Gamma(A)\) form a group (called the ``algebraic cohomotopy group'') under the operation \([a,b]*[c,d]=[ac+de,bc+df]\), where \(e, f\) are such that the matrix with first row \((a,e)\) and second row \((b,f)\) is in \(\mathrm{SL}_2(A)\); the unit element is \([1,0]\). This group was previously used by \textit{M. I. Krusemeyer} in [Invent. Math. 19, 15--47 (1973; Zbl 0247.14005)] (see also [\textit{M. Karoubi} and \textit{O. Villamayor}, Math. Scand. 28, 265--307 (1971; Zbl 0231.18018)]).
The authors establish some properties of the group \(\Gamma(A)\), such as the following: The group is trivial in a number of important cases, namely if \(A\) is a field, a local ring, a polynomial ring over a field, a Euclidean domain, the polynomial ring over integers, semilocal ring, ring of dimension \(0\). If \(I\) is the nil radical of \(A\), then \(\Gamma(A)\cong\Gamma(A/I)\). If \(A=A_0\oplus A_1\oplus A_2\oplus\dots\) is a positively graded ring, then \(\Gamma(A)\cong\Gamma(A_0)\), hence \(\Gamma(A)\cong\Gamma(A[X_1,\dots,X_n])\). If \(K\) is a field, then, for every \(n\), \(\Gamma( K[X_1^{\pm1},\dots, X_n^{\pm1}])=[1,0]\). One of the open problems is whether \(\Gamma(A)\) is torsion-free. The paper would benefit from some linguistic editing.
Reviewer: Radoslav M. Dimitrić (New York)Hypersurface arrangements of aCM typehttps://zbmath.org/1521.130162023-11-13T18:48:18.785376Z"Ballico, Edoardo"https://zbmath.org/authors/?q=ai:ballico.edoardo"Huh, Sukmoon"https://zbmath.org/authors/?q=ai:huh.sukmoonSummary: We investigate the arrangement of hypersurfaces on a nonsingular varieties whose associated logarithmic vector bundle is arithmetically Cohen-Macaulay (for short, aCM), and prove that the projective space is the only smooth complete intersection with Picard rank one that admits an aCM logarithmic vector bundle. We also obtain a number of results on aCM logarithmic vector bundles over several specific varieties. As an opposite situation we investigate the Torelli-type problem that the logarithmic cohomology determines the arrangement.
{{\copyright} 2022 Wiley-VCH GmbH}On the reduction numbers and the Castelnuovo-Mumford regularity of projective monomial curveshttps://zbmath.org/1521.130192023-11-13T18:48:18.785376Z"Lam, Tran Thi Gia"https://zbmath.org/authors/?q=ai:lam.tran-thi-giaLet \(R\) be the coordinate ring of a projective monomial curve given parametrically by a set \(M\) of monomials of degree \(d\) in two variables \(x, y\). Let \(Q=(x ,y )\) be the ideal generated by \(x ,y\) in R. It is known that \(Q\) is a minimal reduction of \(R_+\). In various samples of \(M\) the author computes the minimal reduction number \(r_Q(R_+)\) in terms of \(M\). The approach uses the fact that \(R\) is a finitely generated module over the polynomial ring \(k[x^d,y^d]\), a Noether normalization. Note that the minimal reduction number equals the Castelnuovo-Mumford regularity if \(R\) is a Cohen-Macaulay or more general a Buchsbaum ring.
Reviewer: Peter Schenzel (Halle)Canonical modules and class groups of Rees-like algebrashttps://zbmath.org/1521.130202023-11-13T18:48:18.785376Z"Mantero, Paolo"https://zbmath.org/authors/?q=ai:mantero.paolo"McCullough, Jason"https://zbmath.org/authors/?q=ai:mccullough.jason"Miller, Lance Edward"https://zbmath.org/authors/?q=ai:miller.lance-edwardLet \(S = k[x_1,\ldots,x_n]\) the polynomial ring in \(n\) variables over the field \(k\). For a homogeneous ideal \(I = (f_1,\ldots,f_m)\) the Rees-like algebra is defined by \(S[It, t^2] \subset S[t]\), where \(t\) is a variable. Let \(T = S[y_1,\ldots,y_m,z]\) be the non-standard graded polynomial ring with a natural map \(T \to S[It, t^2]\), where the grading is given by \(\deg y_i = deg f_i +1, \deg z = 2\). Let \(Q\) denote its kernel. Then there is an expression of the canonical module \(\omega_{T/Q} = \operatorname{Ext}^c_T(T/Q,T), c = \operatorname{codim} Q\). Via a concrete computation based on linkage, the authors provide an explicit, well-structured resolution of the canonical module in terms of a type of double-Koszul complex. Additionally, they give descriptions of both the divisor class group and the Picard group of a Rees-like algebra. Note that Rees-like algebra were introduced by Peeva and the second author (see [\textit{J. McCullough} and \textit{I. Peeva}, J. Am. Math. Soc. 31, No. 2, 473--496 (2018; Zbl 1390.13043)])
Reviewer: Peter Schenzel (Halle)Indecomposability of top local cohomology modules and connectedness of the prime divisors graphshttps://zbmath.org/1521.130252023-11-13T18:48:18.785376Z"Doustimehr, Mohammad Reza"https://zbmath.org/authors/?q=ai:doustimehr.mohammad-rezaLet \(R\) be a commutative Noetherian local ring with identity, and let \(\mathfrak a\) be an ideal of \(R\). The goal is to decompose the top local cohomology module \(\text{H}_{\mathfrak a}^d(R)\), where \(d=\dim R\), into indecomposable modules. It is assumed that \(\text{H}_{\mathfrak a}^d(R)\neq 0\) and a natural number \(t\) is given. The main result of this paper states that \(\text{H}_{\mathfrak a}^d(R)\) can be expressed as the direct sum of at least \(t\) nonzero modules if and only if the undirected graph \(\Gamma_{{\mathfrak a},R}\) has at least \(t\) connected components. This result extends the corresponding finding of \textit{M. Hochster} and \textit{C. Huneke} [Contemp. Math. 159, 197--208 (1994; Zbl 0809.13003)]. The vertex set of \(\Gamma_{{\mathfrak a},R}\) consists of all associated prime ideals \({\mathfrak p}\in \text{Ass} \ R\) such that \(\text{H}_{\mathfrak a}^d(R/{\mathfrak p}) \neq 0\), and an edge is formed between any two distinct vertices \(\mathfrak p\) and \(\mathfrak q\) if \(\text{ht}({\mathfrak p}+{\mathfrak q})=1\).
Reviewer: Kamran Divaani-Aazar (Tehran)A cluster structure on the coordinate ring of partial flag varietieshttps://zbmath.org/1521.130342023-11-13T18:48:18.785376Z"Kadhem, Fayadh"https://zbmath.org/authors/?q=ai:kadhem.fayadhSummary: The main goal of this paper is to show that the (multi-homogeneous) coordinate ring of a partial flag variety \(\mathbb{C} [G / P_K^-]\) contains a cluster algebra if \(G\) is any semisimple complex algebraic group. We use derivation properties and a special lifting map to prove that the cluster algebra structure \(\mathcal{A}\) of the coordinate ring \(\mathbb{C} [N_K]\) of a Schubert cell constructed by Goodearl and Yakimov can be lifted, in an explicit way, to a cluster structure \(\widehat{\mathcal{A}}\) living in the coordinate ring of the corresponding partial flag variety. Then we use a minimality condition to prove that the cluster algebra \(\widehat{\mathcal{A}}\) is equal to \(\mathbb{C} [G / P_K^-]\) after localizing some special minors.Algebraic curves and surfaces. A history of shapeshttps://zbmath.org/1521.140012023-11-13T18:48:18.785376Z"Busé, Laurent"https://zbmath.org/authors/?q=ai:buse.laurent"Catanese, Fabrizio"https://zbmath.org/authors/?q=ai:catanese.fabrizio"Postinghel, Elisa"https://zbmath.org/authors/?q=ai:postinghel.elisaThe book contains the notes of three courses given by the authors during the summer school for young researchers TiME2019, held in Levico Terme (Italy).
In the first section, the second author outlines the classification of projective surfaces \(S\), described by Castelnuovo and Enriques at the beginning of the XXth century. The classification is based on the Kodaira dimension \(\kappa(S)\), which corresponds to the dimension of the image of the map associated with \(mK_s\) for \(m\gg 0\). The case \(\kappa(S)=2\) corresponds to surfaces of general type, whose moduli spaces are still intensively studied. The case of rational and ruled surfaces is studied in the first two lectures. Then, the exposition focuses on the \(P_{12}\)-Theorem of Castelnuovo-Enriques, which shows how the birational structure of \(S\) depends on the dimension of the linear system \(12K_S\). In particular, when \(P_{12}>1\) and \(K_S^2=0\), the surface has a fibration \(S\to B\) over a curve, and the fibers are smooth elliptic curves. The classification of this case is based on the isotriviality of the fibration, i.e. the property that the elliptic fibers are isomorphic or, equivalently, their moduli are trivial. In the last lecture, several strategies for the proof of isotriviality are collected and illustrated.
In the second section of the book the third author illustrates the main techniques and some recent achievements in the study of polynomial interpolation with multiplicities. The problem is to determine the dimension of linear systems of polynomials vanishing at \(k\) general points with multiplicities greater or equal than preassigned values. Even in the case of homogeneous polynomials in three variables over the complex field (corresponding to curves in the complex projective plane) the situation is not totally understood. A general method for the construction of linear systems whose dimension is bigger than the expected value is known; it is based on the geometry of the blow up of the plane at several general points. Long ago it has been conjectured that the method exhausts all the cases in which the dimension is bigger than expected. The conjecture is still open. The book contains a discussion on the conjecture, with a description of the cases in which it is known to hold, and an illustration of the main methods used to attack the problem (as Cremona transformations, and the study of the nef cone of blow up's). Possible extensions of the conjecture to higher dimensional spaces are also presented.
The third section contains an outline of algebraic and geometric methods for the computation of implicit (polynomial) equations that describe algebraic varieties defined by parametric polynomials (hence defined as the image of suitable polynomial maps). The construction of implicit equations can provide a way to solve some problems, like the membership problem, relevant for applications. The straightforward method to obtain implicit equations is based on the classical elimination theory. Yet, when the number of variables and parameters increases, a direct use of elimination theory becomes unpractical. The first author describes a series of algebraic tools that can simplify the problem. These tools are based on a matricial representation of the associated ideals, and the author introduces and explains the role of the corresponding elimination matrices. The main application discussed in the section concerns implicit equations that define the intersection of parametric (rational) hypersurfaces, a problem that arises naturally in the reconstruction of images.
Reviewer: Luca Chiantini (Siena)Deformation and unobstructedness of determinantal schemeshttps://zbmath.org/1521.140022023-11-13T18:48:18.785376Z"Kleppe, Jan O."https://zbmath.org/authors/?q=ai:kleppe.jan-oddvar"Miró-Roig, Rosa M."https://zbmath.org/authors/?q=ai:miro-roig.rosa-mariaMotivated by classical projective varieties cut out by minors of a homogeneous matrix such as Veronese varieties, Segre varieties, and rational normal curves, a subscheme \(X \subset \mathbb P^n_k\) of codimension \(c\) is \textit{determinantal} if its homogeneous ideal \(I(X) \subset R = k[x_0, \dots, x_n]\) is the ideal \(I_s (\mathcal A)\) generated by the \(s \times s\) minors of a homogeneous \(p \times q\) matrix \(\mathcal A\) and \(c=(p-s+1)(q-s+1)\). A determinantal scheme \(X \subset \mathbb P^n_k\) is locally Cohen-Macaulay [\textit{M. Hochster} and \textit{J. A. Eagon}, Am. J. Math. 93, 1020--1058 (1971; Zbl 0244.13012)] and even Arithmetically Cohen-Macaulay (ACM). To study families of determinantal schemes, fix integers \(a_1 \leq a_2 \leq \dots \leq a_{t+c-1}\) and \(b_1 \leq b_2 \leq \dots \leq b_t\) such that \(b_i \leq a_i\) for all \(i\) and \(b_{i_0} < a_{i_0}\) for some \(i_0\), and define \(W(\underline b; \underline a; r) \subset \mathrm{Hilb}^{p_X (t)} (\mathbb P^n)\) to be the locus of determinantal schemes \(X \subset \mathbb P^n\) of codimension \(c = r(r+c-1)\) whose ideal is generated by the \((t-r+1) \times (t-r+1)\) minors of a \(t \times (t+c-1)\) matrix \(\mathcal A = (a_{i,j})\), where \(a_{i,j}\) is a homogeneous form of degree \(a_j - b_i\): note that if \(\mathcal A\) drops rank in the expected codimension \(c\), then there is a standard resolution for the ideal \(I_s (\mathcal A)\) given by \textit{A. Lascoux} [Adv. Math. 30, 202--237 (1978; Zbl 0394.14022)], a generalization of the Eagon-Northcott complex [\textit{J. A. Eagon} and \textit{D. G. Northcott}, Proc. R. Soc. Lond., Ser. A 269, 188--204 (1962; Zbl 0106.25603)], which determines the Hilbert polynomial \(p_X (t)\). The locus \(W(\underline b; \underline a; r)\) is irreducible and the conditions on \(a_i, b_i\) assure that it is nonempty. The authors pose the following problems.
\begin{itemize}
\item[(a)] Compute the dimension of \(W(\underline b; \underline a; r)\).
\item[(b)] Determine if \(\overline{W(\underline b; \underline a; r)}\) is an irreducible component of \(\mathrm{Hilb}^{p_X (t)} (\mathbb P^n)\).
\item[(c)] Determine if \(\mathrm{Hilb}^{p_X (t)} (\mathbb P^n)\) is generically smooth along \(W(\underline b; \underline a; r)\).
\item[(d)] Determine if any deformation of \(X \in W(\underline b; \underline a; r)\) comes from deforming the associated homogeneous matrix \(\mathcal A\).
\end{itemize}
\textit{Ellingsrud} achieved ideal results for the case \(c=2, r=1\) by solving all four problems, even showing that \(\mathrm{Hilb}^{p_X (t)} (\mathbb P^n)\) is smooth along \(W(\underline b; \underline a; r)\) [\textit{G. Ellingsrud}, Ann. Sci. Éc. Norm. Supér. (4) 8, 423--431 (1975; Zbl 0325.14002)]. The goal of this work is to extend Ellingsrud's results to arbitrary families of determinantal schemes, to carry out (a) - (d) for any \(r,c,n\) to the extent possible. This program has been partially carried out for determinantal schemes of small codimension when \(r=1\) (\textit{standard} determinantal schemes cut out by maximal minors), for example when \(c=3\) [\textit{J. O. Kleppe} et al., Gorenstein liaison, complete intersection liaison invariants and unobstructedness. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 1006.14018)] or \(c=4\) [\textit{J. O. Kleppe} and \textit{R. M. Miró-Roig}, Trans. Am. Math. Soc. 357, No. 7, 2871--2907 (2005; Zbl 1073.14063)].
The authors' strategy appears in their study of subschemes \(X \subset \mathbb P^n\) of codimension \(c=4\) cut out by submaximal minors of a square matrix [\textit{J. O. Kleppe} and \textit{R. M. Miró-Roig}, Algebra Number Theory 3, No. 4, 367--392 (2009; Zbl 1186.14052)]. They prove their results by considering the Hilbert flag scheme of chains of closed subschemes obtained by deleting suitable columns of the associated matrices and the projections to the usual Hilbert schemes. Under numerical conditions they prove recursively that the closure of \(W(\underline b, \underline a, r)\) is a generically smooth component of \(\mathrm{Hilb}^{p_X (t)} (\mathbb P^n)\), starting from the induction base \(W(\underline b, \underline a, 1)\) parametrizing standard determinantal schemes. Their proof is based on (1) a close analysis as to whether any deformation of a determinantal scheme \(X\) in \(W(\underline b, \underline a,r)\) comes from deforming its associated matrix and (2) the interesting observation that any determinantal scheme \(X\) is defined by a regular section of a sheaf on a determinantal scheme \(Y\) of lower codimension. To explain point (2), let \(\varphi: \oplus_{i=1}^t R(b_i) \to \oplus_{j=1}^{t+c-1}\) be the map defined by \(\mathcal A^T\) and \(\phi: \oplus_{i=1}^t R(b_i) \to \oplus_{j=1}^{t+c-2}\) the map defined by deleting the last column of \(\mathcal A\). Let \(X = \mathrm{Proj} \; A\) with \(A = R/I_{t+1-r} (\varphi^*)\) and \(Y = \mathrm{Proj} \; B\) with \(B = R/I_{t+1-r} (\phi^*)\). Then if \(M = \mathrm{coker} (\varphi^*)\) and \(N = \mathrm{coker} \phi^*\), they prove that there is an exact sequence \[ 0 \to B(-a_{t+c-1}) \stackrel{\sigma^*}{\rightarrow} N \otimes B \to M \otimes B \to 0\tag{1} \] defining a regular section \(\sigma^*\) as shown such that \(A = B/\mathrm{im}(\sigma)\) provided that \(\dim Y > r\) and \(Y\) is regular in codimension one. This extends to the case when \(A\) is artinian and \(k\) is replaced by a local artinian ring, which is used to show that any deformation of \(X\) comes from a deformation of the matrix \(\mathcal A\). Combined with results of \textit{W. Bruns} [Compos. Math. 47, 171--193 (1982; Zbl 0506.13007)], they deduce the vanishing of various \(\mathrm{Ext}^i_B\) groups involving \(N \otimes B, M, B\) and \(I_{A/B} = \mathrm{ker} (B \to A)\), which give the authors a foothold from which to prove their results.
We briefly summarize the 10 chapters and appendix. Chapters 2 and 3 give background and mild generalizations of known results.
Chapter 4 develops exact sequence (1), the module \(N\), and proves the vanishing of certain \(\mathrm{Ext}\) groups. In particular, the vanishing \(\mathrm{Ext}^1_A (I_{A/B}/I^2_{A/B},A)=0\) proved under a depth condition on \(A\) shows that the second projection from the Hilbert scheme \(\mathrm{Hilb}^{p_X (t), p_Y (t)} (\mathbb P^n)\) is smooth at \((X \subset Y)\).
Chapter 5 characterizes when any deformation of the determinantal ring \(A\) comes from deforming its associated homogeneous matrix \(\mathcal A\) in terms of surjectivity of its tangent map. This holds for generic determinantal rings provided \((s,c) \neq (t,1)\), allowing the authors to prove that if any deformation of \(B\) comes from deforming its associated matrix and \({}_0\mathrm{Ext}^1_B(I_B/I_B^2,I_{A/B})=0\), then the same holds for \(A\). This implies that the first projection from the flag Hilbert scheme \(\mathrm{Hilb}^{p_X (t), p_Y (t)} (\mathbb P^n)\) is smooth at \((X \subset Y)\) with tangential fiber dimension \(\dim {}_0\mathrm{Hom}_B(I_B/I_B^2,I_{A/B})\).
Chapter 6 gives an upper bound on the dimension of the family \(W(\underline b, \underline a, r)\) independent of \(r\) and numerical conditions under which equality holds. The authors give a conjectural formula for \(\dim {}_0\mathrm{Hom}_B(I_B/I_B^2,I_{A/B})\) when \(\dim B > r+2\), which leads to an exact formula for the dimension of \(W(\underline b, \underline a, r)\). They also give counterexamples when \(\dim B = r+2\) and \(r=2\).
Chapter 7 is devoted to proving statements (b) and (c) of the program, mainly using (d) to see when \(\overline{W(\underline b, \underline a, r)}\) is a generically smooth component of \(\mathrm{Hilb}^{p_X (t)} (\mathbb P^n)\). The main result gives various conditions under which the latter holds, but for more flexible applications the authors assume instead of condition (d) that every deformation of \(B\) comes from deformations of the corresponding matrix and that the natural map \(\gamma: {}_0 \mathrm{Hom}_R (I_A,A) \to {}_0 Ext^1_B (I_B/I_B^2,I_{A/B})\) is zero, a condition amenable to calculation with Macaulay2 [\url{http://www.math.uiuc.edu/Macaulay2/}]. While the hypotheses are technical, many special cases and examples are given. The authors conjecture that problems (b) and (c) have positive answers when \(\dim A \geq 4\) for \(c=1\) (\(\dim A \geq 3\) for \(c \neq 1\)) and \(a_1 > b_t\).
Chapter 8 weakens the hypotheses of the results in chapter 6 and chapter 7: those results have assumptions \({}_0 \mathrm{Ext}^1_B (I_B/I_B^2,I_{A/B})=0\) (or \(\gamma =0\)) and conditions on \(\dim \mathrm{Hom}_B (I_B/I_B^2,I_{A/B})\) and \(\dim (M \otimes A)_{(a_{t+c-1})}\) to compute the dimension of \(\overline{W(\underline b, \underline a, r)}\). To remove these, the authors work directly with flags of determinantal rings \(A_{2-r} \to A_{3-r} \to \dots \to A_0 \to A_1 \to \dots \to A_c = A\) obtained by removing one column at a time from the associated matrices, with the goal of solving the main problems by recursively transfering the property that any deformation of a matrix comes from deforming its associated homogeneous matrix from one ring \(A_j\) to the next \(A_{j+1}\) starting at \(j=2-r\), the standard determinantal setting. This idea works only starting at \(j=3-r\), so they have to build the transfer from \(j=r-2\) to \(j=r-3\) into the hypothesis of their results.
Chapter 9 gives isomorphisms between (a) the deformation functor of \(\wedge^r M\) as an \(R\)-module, (b) the deformation functor of the surjection \(R \to A\), and (c) the local Hilbert functor of \(X\) under a depth condition on \(A\). With more hypotheses they use this to conclude that for all \(i\), \(\overline{W(\underline b, \underline a, i)} \subset \mathrm{Hilb}^{p_{X_i} (t)} (\mathbb P^n)\) is a generically smooth irreducible component of the same dimension because the corresponding local Hilbert functors are isomorphic, even though their Hilbert polynomials are very different.
Chapter 10 collects several leftover open questions and conjectures based on the work presented. The paper is full of examples, some coming from the results and others computed with Macaulay2 [loc. cit.], so the authors include an appendix showing their source code for those calculations.
Reviewer: Scott Nollet (Fort Worth)Mapping stacks and categorical notions of propernesshttps://zbmath.org/1521.140032023-11-13T18:48:18.785376Z"Halpern-Leistner, Daniel"https://zbmath.org/authors/?q=ai:halpern-leistner.daniel"Preygel, Anatoly"https://zbmath.org/authors/?q=ai:preygel.anatolySummary: One fundamental consequence of a scheme \(X\) being proper is that the functor classifying maps from \(X\) to any other suitably nice scheme or algebraic stack is representable by an algebraic stack. This result has been generalized by replacing \(X\) with a proper algebraic stack. We show, however, that it also holds when \(X\) is replaced by many examples of algebraic stacks which are not proper, including many global quotient stacks. This leads us to revisit the definition of properness for stacks. We introduce the notion of a formally proper morphism of stacks and study its properties. We develop methods for establishing formal properness in a large class of examples. Along the way, we prove strong \(h\)-descent results which hold in the setting of derived algebraic geometry but not in classical algebraic geometry. Our main applications are algebraicity results for mapping stacks and the stack of coherent sheaves on a flat and formally proper stack.Algebraic spaces that become schematic after ground field extensionhttps://zbmath.org/1521.140042023-11-13T18:48:18.785376Z"Schröer, Stefan"https://zbmath.org/authors/?q=ai:schroer.stefanSummary: We construct examples of non-schematic algebraic spaces that become schemes after finite ground field extensions.
{{\copyright} 2022 The Authors. \textit{Mathematische Nachrichten} published by Wiley-VCH GmbH.}Representation homology of simply connected spaceshttps://zbmath.org/1521.140052023-11-13T18:48:18.785376Z"Berest, Yuri"https://zbmath.org/authors/?q=ai:berest.yuri-yu"Ramadoss, Ajay C."https://zbmath.org/authors/?q=ai:ramadoss.ajay-c"Yeung, Wai-Kit"https://zbmath.org/authors/?q=ai:yeung.wai-kitSummary: Let \(G\) be an affine algebraic group defined over a field \(k\) of characteristic 0. We study the derived moduli space of \(G\)-local systems on a pointed connected CW complex \(X\) trivialized at the basepoint of \(X\). This derived moduli space is represented by an affine DG scheme \(\mathbf{R}\operatorname{Loc}_G(X,\ast )\): we call the (co)homology of the structure sheaf of \(\mathbf{R}\operatorname{Loc}_G(X,\ast )\) the \textit{representation homology of} \(X\) in \(G\) and denote it by \(\mathrm{HR}_\ast (X,G)\). The 0-dimensional homology, \( \mathrm{HR}_0(X,G)\), is isomorphic to the coordinate ring of the \(G\)-representation variety \({\mathrm{Rep}}_G[\pi_1(X)]\) of the fundamental group of \(X\) -- a well-known algebro-geometric invariant that plays a role in many areas of topology. The higher representation homology is much less studied. In particular, when \(X\) is simply connected, \( \mathrm{HR}_0(X,G)\) is trivial but \(\mathrm{HR}_*(X,G)\) is still an interesting rational invariant of \(X\) that depends on the Lie algebra of \(G\). In this paper, we use Quillen's rational homotopy theory to compute the representation homology of an arbitrary simply connected space (of finite rational type) in terms of its Lie and Sullivan algebraic models. When \(G\) is reductive, we also compute \(\mathrm{HR}_\ast (X,G)^G\), the \(G\)-invariant part of representation homology, and study the question when \(\mathrm{HR}_\ast (X,G)^G\) is free of locally finite type as a graded commutative algebra. This question turns out to be related to the so-called Strong Macdonald Conjecture, a celebrated result in representation theory proposed (as a conjecture) by Feigin and Hanlon in the 1980s and proved by \textit{S. Fishel} et al. in [Ann. Math. (2) 168, No. 1, 175--220 (2008; Zbl 1186.17010)]. Reformulating the Strong Macdonald Conjecture in topological terms, we give a simple characterization of spaces \(X\) for which \(\mathrm{HR}_\ast (X,G)^G\) is a graded symmetric algebra for any complex reductive group \(G\).
{{\copyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}A new family of isolated symplectic singularities with trivial local fundamental grouphttps://zbmath.org/1521.140062023-11-13T18:48:18.785376Z"Bellamy, Gwyn"https://zbmath.org/authors/?q=ai:bellamy.gwyn"Bonnafé, Cédric"https://zbmath.org/authors/?q=ai:bonnafe.cedric"Fu, Baohua"https://zbmath.org/authors/?q=ai:fu.baohua"Juteau, Daniel"https://zbmath.org/authors/?q=ai:juteau.daniel"Levy, Paul"https://zbmath.org/authors/?q=ai:levy.paul-d"Sommers, Eric"https://zbmath.org/authors/?q=ai:sommers.eric-nThe introduction by Beauville of symplectic singularities has led to numerous important developments in both algebraic geometry and geometric representation theory. Basic examples of symplectic singularities include symplectic quotient singularities and singularities in normalizations of nilpotent orbit closures in semi-simple Lie algebras.
An isolated symplectic singularity of dimension at most 4 is a normal isolated singularity whose smooth locus admits a holomorphic symplectic 2-form. It follows that such a singularity is canonical Gorenstein, hence its local fundamental group is finite. Symplectic quotient singularities give many examples of isolated symplectic singularities, but they all have non-trivial local fundamental groups. The authors construct a new infinite family of four-dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as singularities in blowups of the quotient of \(\mathbb{C}^4\) by the dihedral group of order \(2d\), (2) as singular points of Calogero-Moser spaces associated with dihedral groups of order \(2d\) at equal parameters, and (3) as singularities of a certain Slodowy slice in the \(d\)-fold cover of the nilpotent cone in \(sl_d\).
Reviewer: Vladimir P. Kostov (Nice)ACC conjecture for the weighted log canonical thresholds in dimension twohttps://zbmath.org/1521.140072023-11-13T18:48:18.785376Z"Nguyen Xuan Hong"https://zbmath.org/authors/?q=ai:nguyen-xuan-hong.The weighted log canonical threshold or weighted complex singularity \(c_t(f)\) exponent of a holomorphic function \(f\) in a neighbourhood of the origin in \(\mathbb C^n\) is the supremum of constants such that \(||z||^{2t}|f|^{-2c}\) is integrable. For the unweighted case (\(t=0\)) the set of possible values is known to satisfy the Ascending Chain Condition. \textit{N. X. Hong} et al. [Anal. Math. 46, No. 1, 77--83 (2020; Zbl 1449.14002)] proved the ACC conjecture the weighted case for \(n=2\) and \(t>0\). The author completes the proof by showing that ACC also holds for \(-2<t<0\).
Reviewer: Jan Stevens (Göteborg)Explicit minimal embedded resolutions of divisors on models of the projective linehttps://zbmath.org/1521.140082023-11-13T18:48:18.785376Z"Obus, Andrew"https://zbmath.org/authors/?q=ai:obus.andrew"Srinivasan, Padmavathi"https://zbmath.org/authors/?q=ai:srinivasan.padmavathiSummary: Let \(K\) be a discretely valued field with ring of integers \(\mathcal{O}_K\) with perfect residue field. Let \(K(x)\) be the rational function field in one variable. Let \(\mathbb{P}^1_{\mathcal{O}_K}\) be the standard smooth model of \(\mathbb{P}^1_K\) with coordinate \(x\). Let \(f(x)\in\mathcal{O}_K[x]\) be a squarefree polynomial with corresponding divisor of zeroes \(\operatorname{div}_0(f)\) on \(\mathbb{P}^1_{\mathcal{O}_K}\). We give an explicit description of the minimal embedded resolution \(\mathcal{Y}\) of the pair \((\mathbb{P}^1_{\mathcal{O}_K}, \operatorname{div}_0(f))\) by using Mac Lane's theory to write down the discrete valuations on \(K(x)\) corresponding to the irreducible components of the special fiber of \(\mathcal{Y}\).The delta invariant and fiberwise normalization for families of isolated non-normal singularitieshttps://zbmath.org/1521.140092023-11-13T18:48:18.785376Z"Greuel, Gert-Martin"https://zbmath.org/authors/?q=ai:greuel.gert-martin"Pfister, Gerhard"https://zbmath.org/authors/?q=ai:pfister.gerhardThe delta invariant, also called genus defect, is an important numerical invariant of a singular reduced curve and is therefore often considered for algebraic curves over the complex numbers, but also for curves over finite fields, e.g. in coding theory. The delta invariant has been extended to generically reduced complex analytic curves and it has been shown that it can be used to control the topology in a family of such curves by taking care of the influence of embedded points.
This invariant has been further extended to complex-analytic isolated non-normal singularities of any dimension and its behavior has been studied in connection with simultaneous normalization.
The authors prove the semicontinuity of the delta invariant in a family of schemes or analytic varieties with finitely many (not necessarily reduced) isolated non-normal singularities, in particular for families of generically reduced curves. They define and use a modified delta invariant for isolated non-normal singularities of any dimension that takes care of embedded points. Their results generalize results by Teissier and Chiang-Hsieh-Lipman for families of reduced curve singularities.
The base ring for the families can be an arbitrary principal ideal domain such that the semicontinuity result provides possible improvements for algorithms to compute the genus of a curve.
Reviewer: Vladimir P. Kostov (Nice)Ramification theory of reciprocity sheaves. II: Higher local symbolshttps://zbmath.org/1521.140102023-11-13T18:48:18.785376Z"Rülling, Kay"https://zbmath.org/authors/?q=ai:rulling.kay"Saito, Shuji"https://zbmath.org/authors/?q=ai:saito.shuji.1Summary: We construct a theory of higher local symbols along Paršin chains for reciprocity sheaves. Applying this formalism to differential forms, gives a new construction of the Paršin-Lomadze residue maps, and applying it to the torsion characters of the fundamental group gives back the reciprocity map from Kato's higher local class field theory in the geometric case. The higher local symbols satisfy various reciprocity laws. The main result of the paper is a characterization of the modulus attached to a section of a reciprocity sheaf in terms of the higher local symbols.
For Part I, see [\textit{K. Rülling} and \textit{S. Saito}, J. Reine Angew. Math. 797, 41--78 (2023; Zbl 07669616)].Nearby motivic sheaves of weighted equivariant functionshttps://zbmath.org/1521.140112023-11-13T18:48:18.785376Z"Ivorra, Florian"https://zbmath.org/authors/?q=ai:ivorra.florian"Sebag, Julien"https://zbmath.org/authors/?q=ai:sebag.julienSummary: Let \(k\) be a field of characteristic zero and let \(X\) be an algebraic \(k\)-variety endowed with an action of the multiplicative algebraic monoid \(\mathbf{A}^1\). In this article, we prove that the nearby motivic sheaf functor of a weighted \(\mathbf{A}^1\)-equivariant function on \(X\) commutes with direct (and exceptional direct) images when applied to constant objects and their twists by Thom spaces of equivariant vector bundles. This proves a generalized categorified version of the conjectures of \textit{K. Behrend} et al. [Invent. Math. 192, No. 1, 111--160 (2013; Zbl 1267.14008)] and \textit{B. Davison} and \textit{S. Meinhardt} [Geom. Topol. 19, No. 5, 2535--2555 (2015; Zbl 1430.14105)] in the context of motivic stable homotopy theory. In particular, our formula is functorial, holds for general weighted \(\mathbf{A}^1\)-equivariant functions and provides, as a direct consequence, a positive answer to the conjectures of Behrend-Bryan-Szendrői and Davison-Meinhardt for virtual motives.On the motive of the nested Quot scheme of points on a curvehttps://zbmath.org/1521.140122023-11-13T18:48:18.785376Z"Monavari, Sergej"https://zbmath.org/authors/?q=ai:monavari.sergej"Ricolfi, Andrea T."https://zbmath.org/authors/?q=ai:ricolfi.andrea-tLet \(K_0(Var_k)\) be the Grothendieck ring of varieties over an algebraically closed field \(k\). Let \(Y\) be a \(k\)-variety. The motivic zeta function of \(Y\) is defined as :
\[
\zeta_Y(q)=1+\sum_{n>0}[\mathrm{Sym}^nY]q^n
\]
where \(\mathrm{Sym}^n Y\) denote the \(n\)-th symmetric power of \(Y\) and in the above sum we consider the class of the symmetric powers in the Grothendieck ring of varieties over \(k\). This zeta function was introduced by Kapranov, where he proved that for smooth curves it is a rational function of \(q\).
In this paper the authors compute the multivariable zeta function of the nested Quot scheme of points \(\mathrm{Quot}_C(E,\mathbf{n})\) on a smooth curve \(C\) in terms of \(\zeta_C(q)\). Here we fix a locally free sheaf \(E\) on \(C\) and \(\mathbf{n}=(0\leq n_1\leq n_2\cdots\leq n_d)\), \(d\) is fixed. \(\mathrm{Quot}_C(E,\mathbf{n})\) is a flag of quotients of \(E\):
\[
E\rightarrow T_d\cdots \rightarrow T_1
\]
where each \(T_i\) is a zero dimensional sheaf of length \(n_i\). All the arrows in the above are surjective.
The main result of this paper is:
Let \(C\) be a smooth curve defined over an algebraically closed field \(k\) and \(E\) be locally free on \(C\), then:
\[
\sum [\mathrm{Quot}_C(E,\mathbf{n})]q_1^{n_1}\cdots q_d^{n_d}=\prod_{\alpha=1}^r\prod_{i=1}^d \zeta_C(\mathbb L^{\alpha-1}q_iq_{i+1}\cdots q_d)
\]
In particular the above generating function is rational in \(q_1,\cdots,q_d\).
Reviewer: Kalyan Banerjee (Chennai)New elementary components of the Gorenstein locus of the Hilbert scheme of pointshttps://zbmath.org/1521.140132023-11-13T18:48:18.785376Z"Szafarczyk, Robert"https://zbmath.org/authors/?q=ai:szafarczyk.robertSummary: We construct new explicit examples of nonsmoothable Gorenstein algebras with Hilbert function \((1,n,n,1)\). This gives a new infinite family of elementary components in the Gorenstein locus of the Hilbert scheme of points and solves the cubic case of Iarrobino's conjecture.Pushforwards of Chow groups of smooth ample divisors, with an emphasis on Jacobian varietieshttps://zbmath.org/1521.140142023-11-13T18:48:18.785376Z"Banerjee, Kalyan"https://zbmath.org/authors/?q=ai:banerjee.kalyan.1"Iyer, Jaya N. N."https://zbmath.org/authors/?q=ai:iyer.jaya-n-n"Lewis, James D."https://zbmath.org/authors/?q=ai:lewis.james-dSummary: With a homological Lefschetz conjecture in mind, we prove the injectivity of the pushforward morphism on low-dimensional rational Chow groups, induced by the closed embedding of an ample divisor, namely, the Theta divisor inside the Jacobian variety \(J(C)\). Here, \(C\) is a smooth irreducible complex projective curve.
{{\copyright} 2022 Wiley-VCH GmbH.}The Chow rings of moduli spaces of elliptic surfaces over \(\mathbb{P}^1\)https://zbmath.org/1521.140152023-11-13T18:48:18.785376Z"Canning, Samir"https://zbmath.org/authors/?q=ai:canning.samir"Kong, Bochao"https://zbmath.org/authors/?q=ai:kong.bochaoSummary: Let \(E_N\) denote the coarse moduli space of smooth elliptic surfaces over \(\mathbb{P}^1\) with fundamental invariant \(N\). We compute the Chow ring \(A^\ast(E_N)\) for \(N\geqslant 2\). For each \(N\geqslant 2\), \(A^\ast(E_N)\) is Gorenstein with socle in codimension 16, which is surprising in light of the fact that the dimension of \(E_N\) is \(10N-2\). As an application, we show that the maximal dimension of a complete subvariety of \(E_N\) is 16. When \(N=2\), the corresponding elliptic surfaces are \(K3\) surfaces polarized by a hyperbolic lattice \(U\). We show that the generators for \(A^\ast(E_2)\) are tautological classes on the moduli space \(\mathcal{F}_U\) of \(U\)-polarized \(K3\) surfaces, which provides evidence for a conjecture of Oprea and Pandharipande on the tautological rings of moduli spaces of lattice polarized \(K3\) surfaces.Motivic sheaves revisitedhttps://zbmath.org/1521.140162023-11-13T18:48:18.785376Z"Arapura, Donu"https://zbmath.org/authors/?q=ai:arapura.donuSummary: The purpose of this paper is to present a simplified construction of the author's category of motivic sheaves [\textit{D. Arapura}, Adv. Math. 233, No. 1, 135--195 (2013; Zbl 1277.14013)], and to provide a simplified proof of a theorem of \textit{D. Arapura} [Invent. Math. 160, No. 3, 567--589 (2005; Zbl 1083.14011)] that the Leray spectral sequence can be lifted to this category.Fujita's conjecture for quasi-elliptic surfaceshttps://zbmath.org/1521.140172023-11-13T18:48:18.785376Z"Chen, Yen-An"https://zbmath.org/authors/?q=ai:chen.yen-anSummary: We show that Fujita's conjecture is true for quasi-elliptic surfaces. Explicitly, for any quasi-elliptic surface \(X\) and an ample line bundle \(A\) on \(X\), we have \(K_X + t A\) is base point free for \(t \geq 3\) and is very ample for \(t \geq 4\).
{{\copyright} 2021 Wiley-VCH GmbH}Positivity of divisors on blow-up projective spaces. Ihttps://zbmath.org/1521.140182023-11-13T18:48:18.785376Z"Dumitrescu, Olivia"https://zbmath.org/authors/?q=ai:dumitrescu.olivia-m"Postinghel, Elisa"https://zbmath.org/authors/?q=ai:postinghel.elisaIn the paper under review, the authors study positivity of divisors on blown-ups of projective spaces. Let \(\mathbb{K}\) be an algebraically closed field of characteristic zero. Let \(\mathcal{S} = \{p_{1}, ..., p_{s}\}\) be a collection of \(s\) distinct points in \(\mathbb{P}^{n}_{\mathbb{K}}\) and let \(S\) be the set of incidences parametrizing \(\mathcal{S}\) with \(|S| = s\). Let \[\mathcal{L} := \mathcal{L}_{m,d}(m_{1}, ..., m_{s})\] denote the linear system of degree \(d\) hypersurfaces in \(\mathbb{P}^{n}_{\mathbb{K}}\) with multiplicity at least \(m_{i}\) at \(p_{i}\) with \(i \in \{1, ..., s\}\). Denote by \(X_{s}\) the blow-up of \(\mathbb{P}^{n}_{\mathbb{K}}\) in the points of \(\mathcal{S}\) and by \(E_{i}\) the exceptional divisor of \(p_{i}\) for all \(i\). For non-negative integers \(d, m_{1}, ..., m_{s}\) we define the following line bundle on \(X_{s}\) \[(\star) : \quad L = dH - \sum_{i=1}^{s}m_{i}E_{i},\] and we denote by \(D\) a general section of \(L\). Denote by \(s(d) := s_{D}(d)\) the number of points in \(\mathcal{S}\) at which the multiplicity of \(D\) equals to \(d\); this number depends on \(\mathcal{L}\) or, equivalently, on \(D\). We introduce \[b = b(D) := \min\{n-s(d), s-n-2\}.\]
Definition. Let \(X\) be a smooth projective variety. For an integer \(\ell > 0\), a line bundle \(\mathcal{O}_{X}(D)\) on \(X\) is said to be \(\ell\)-very ample, if for every \(0\)-dimensional subscheme \(Z \subset X\) of length \(h^{0}(Z, \mathcal{O}_{Z}) = \ell +1\), the restriction map \[H^{0}(X,\mathcal{O}_{X}(D)) \rightarrow H^{0}(Z,\mathcal{O}_{X}(D)|_{Z})\] is surjective.
Here are the main results of the paper.
Theorem A. Assume that \(\mathcal{S} \subset \mathbb{P}^{n}_{\mathbb{K}}\) is a collection of points in general position. Let \(l\) be a non-negative integer. Assume that either \(s \leq 2n\) or \(s \geq 2n+1\) and \(d\) is large enough, namely \[d > l + 2, \quad \sum_{i=1}^{s}m_{i} -nd \leq b_{l},\] where \(b_{l}\) is defined as follows. For every \(s \geq n+3\) we put \[b_{l} = \min \{n-1,s-n-2\} - l -1\] provided that \(m_{1} = d-l-1\) and \(m_{i}=1\) for \(i \geq 2\), or \(b_{l} =\min\{n,s-n-2\}-l-1\) otherwise, while for \(s \leq n+2\) one defines \(b_{l}=-l-1\).
If \(n=1\), then \(D\) is \(l\)-very ample. If \(n\geq 2\), the divisor \(D\) of the form \((\star)\) is \(l\)-very ample if and only if \[l \leq m_{i}, \quad \forall \, i\in \{1, ..., s\},\] \[l \leq d-m_{i}-m_{j}, \quad \forall \, i,j \in \{1, ..., s\}, i\neq j.\]
Theorem B. In the same notation as in Theorem A, assume that for \(D\) of the form \((\star)\) we have that either \(s \leq 2n\) or \(s \geq 2n+1\) and \[d > l + 2, \quad \sum_{i=1}^{s}m_{i} -nd \leq b_{l}\] with \(l=0\) is satisfied. Then \(D\) is nef if and only if it is globally generated.
Theorem C. Let \(X_{s}\) be the blown-up of \(\mathbb{P}^{n}_{\mathbb{K}}\) at \(s\) points in general position with \(s\leq 2n\). Then the Fujita's conjecture holds for \(X_{s}\).
Reviewer: Piotr Pokora (Kraków)On the numerical dimension of Calabi-Yau 3-folds of Picard number 2https://zbmath.org/1521.140192023-11-13T18:48:18.785376Z"Hoff, Michael"https://zbmath.org/authors/?q=ai:hoff.michael"Stenger, Isabel"https://zbmath.org/authors/?q=ai:stenger.isabelLet \(X\) be a normal variety and \(D\) a divisor on it, the numerical dimension of \(D\) is the numerical version of the Itaka dimension, which play a central role in birational geometry. In this article it's defined as the numerical invariant which measure the asymptotic growth of \(h^0(X, mD+A)\) for increasing \(m\) with \(A\) an ample divisor.
The author of the paper under review prove that for any Calabi-Yau threefold \(X\) with Picard number \(2\) and infinite birational automorphism group, the numerical dimension of any \(\mathbb{R}\)-divisor on the boundary of the movable cone a is equal to \(\dfrac{3}{2}\) (Main Theorem).
As the authors explain in Section \(1\), the proof of the Main Theorem is an adaptation of the one done for a specific example in [\textit{J. Lesieutre}, J. Algebr. Geom. 31, No. 1, 113--126 (2022; Zbl 1484.14015)]. In Section \(2\) the authors prove the key result (Corollary \(2.10.\)) which allows them to apply Lesieutre method in a more general setting. This result concerns the description of the fundamental domain for the action of birational automorphism group on the movable cone. Then, in Section \(3\) they prove the Main Theorem.
Finally, Section \(4\) is devoted to the construction of new Calabi-Yau threefolds of Picard number \(2\) with infinite birational automorphism group. In particular, on these Calabi-Yau threefolds there exist birational involutions. These examples are given as complete intersections on a fivefold of Picard number \(2\) isomorphic to the projectivization of the universal rank \(2\) quotient bundle on the Grassmannian \(G(2,4) \subset \mathbb{P}^5\).
Reviewer: Martina Monti (Milano)Torsion codimension 2 cycles on supersingular abelian varietieshttps://zbmath.org/1521.140202023-11-13T18:48:18.785376Z"Gregory, Oliver"https://zbmath.org/authors/?q=ai:gregory.oliverThis paper is concerned with the convergence of empirical spectral distributions (ESD) of random matrices, both in the sense of convergence in probability and in the almost sure sense.
This paper is concerned with Griffiths groups of abelian threefolds over finite fields. Fix a prime number \(p\), and an algebraically closed field \(k\) of characteristic \(p\). For a smooth projective variety \(X\) over \(k\) and \(i \in \mathbb Z_{\geq 0}\), consider any Weil cohomology theory, and let
\[
\mathrm{CH}^i(X)_{\text{hom}} \subset \mathrm{CH}^i(X)
\]
be the group of codimension \(i\) cycles on \(X\) which are homologous to zero, modulo rational equivalence. This contains the group \(\mathrm{CH}^i(X)_{\text{alg}} \subset \mathrm{CH}^i(X)\) of rational equivalence classes of algebraically trivial codimension \(i\) cycles; define
\[
\mathrm{Griff}^i(X) = \mathrm{CH}^i(X)_{\text{hom}}/ \mathrm{CH}^i(X)_{\text{alg}}.
\]
A priori, the definition of the group \(\mathrm{CH}^i(X)_{\text{hom}}\), and hence of the group \(\mathrm{Griff}^i(X)\), depends on the choice of Weil cohomology theory for \(X\). The standard conjectures (more precisely, the conjecture that homological equivalence coincides with numerical equivalence) would imply that \(\mathrm{CH}^i(X)_{\text{hom}}\) is independent of the choice of Weil cohomology theory.
Let \(A\) be an abelian variety over \(k\). We say that \(A\) is said supersingular if all the slopes of the associated \(p\)-divisible group \(A[p^\infty]\) equal \(1/2\). %Recall that a smooth projective variety X over k is said to be supersingular if the Newton polygons of X are isoclinic.
By a well-known theorem of Oort, \(A\) is supersingular if and only if \(A\) is isogenous to the self-product \(E^g\) of any supersingular elliptic curve over \(k\).
Let \(A\) be a supersingular abelian variety over \(k\). As Gregory remarks in the paper under consideration, a result of Fakhruddin [\textit{N. Fakhruddin}, Can. Math. Bull. 45, No. 2, 204--212 (2002; Zbl 1057.14013)] implies that for each \(i \in \mathbb Z_{\geq 0}\), one has
\[
\mathrm{CH}^i(A)_{\text{hom}} \otimes \mathbb Q = \{ x \in \mathrm{CH}^i(A) \otimes \mathbb Q \mid [m]^\ast(x) = m^{2i - 1} \cdot x \} \subset \mathrm{CH}^i(A) \otimes \mathbb Q.
\]
In particular, the group \(\mathrm{CH}^i(A)_{\text{hom}} \otimes \mathbb Q \) is independent of the chosen Weil cohomology theory, and hence the same holds for the groups \(\mathrm{CH}^i(A)_{\text{hom}}\) and \(\mathrm{Griff}^i(A)\).
\textit{C. Schoen} [Ann. Sci. Éc. Norm. Supér. (4) 28, No. 1, 1--50 (1995; Zbl 0839.14004)] showed that if \(k = \bar{\mathbb F}_p\) and \(p \equiv 2 \bmod 3\) and \(E\) denotes the Fermat cubic, then \(\mathrm{Griff}^2(E^3)\) is at most a \(p\)-primary torsion group. The condition that \(p \equiv 2 \bmod 3\) implies that \(E^3\) is a supersingular abelian threefold. Using the work of Fakhruddin mentioned above, \textit{B. B. Gordon} and \textit{K. Joshi} [Can. Math. Bull. 45, No. 2, 213--219 (2002; Zbl 1056.14011)] generalized Schoen's result to all supersingular abelian varieties. Thus, by then it was known that the codimension \(2\) Griffiths groups of supersingular abelian varieties defined over the algebraic closure of \(\mathbb F_p\) are at most \(p\)-primary torsion. Since then, the question of whether these groups possess nontrivial \(p\)-torsion had remained open.
In the article under consideration, Gregory completes the above results by proving the following theorem.
Theorem 1.1. Let \(k\) be an algebraically closed field of characteristic \(p > 0\), and let \(A\) be an abelian variety over \(k\). Then the inclusions \(\mathrm{CH}^2(A)_{\text{alg,tors}} \subset \mathrm{CH}^2(A)_{\text{hom,tors}} \subset \mathrm{CH}^2(A)_{\text{tors}}\) are equalities.
As a consequence, Gregory obtains:
Theorem 1.2. Let \(p\) be a prime number and \(k = \bar{\mathbb F}_p\). Let \(A\) be a supersingular abelian variety over \(k\). Then \(\textrm{Griff}^2(A)[p^\infty] = 0\).
By combining this result with Theorem 5.1 of [\textit{B. B. Gordon} and \textit{K. Joshi}, Can. Math. Bull. 45, No. 2, 213--219 (2002; Zbl 1056.14011)], the author concludes that \(\mathrm{Griff}^2(A)\) is trivial for \(k = \bar{\mathbb F}_p\) and \(A\) as above. That is, homological equivalence coincides with algebraic equivalence for codimension \(2\) cycles on supersingular abelian varieties over the algebraic closure of a finite field.
Reviewer: Olivier de Gaay Fortman (Hannover)On classifying spaces of spin groupshttps://zbmath.org/1521.140212023-11-13T18:48:18.785376Z"Karpenko, Nikita A."https://zbmath.org/authors/?q=ai:karpenko.nikita-aSummary: For a split maximal torus \(T\) of a split spin group \(G=\mathrm{Spin}(n)\) over an arbitrary field, we consider the restriction homomorphism \(f:\mathrm{CH}(BG) \to \mathrm{CH} (BT)^W\) of the Chow rings of their classifying spaces with \(W\) the Weyl group of \(G\). For \(n\leq 6, f\) is known to be surjective. For \(n\geq 7\), an obstruction for an element of \(\mathrm{CH} (BT)^W\) to be in the image of \(f\) is given by the Steenrod operations on \(\mathrm{CH} (BT)/2 \mathrm{CH}(BT)\). Using it, we show that several standard generators of \(\mathrm{CH} (BT)^W\), including the defined for even \(n\) Euler class \(e \in \mathrm{CH}^{n/2} (BT)^W\), are outside the image of \(f\). This result differs from the analogues topological result.\( \mathfrak{sl}(2)\)-type singular fibres of the symplectic and odd orthogonal Hitchin systemhttps://zbmath.org/1521.140222023-11-13T18:48:18.785376Z"Horn, Johannes"https://zbmath.org/authors/?q=ai:horn.johannesThe Hitchin fibration played a major role in two recent developments in the theory of Higgs bundle moduli spaces: First, in the study of the asymptotic of the hyperkähler metric [\textit{R. Mazzeo} et al., Commun. Math. Phys. 367, No. 1, 151--191 (2019; Zbl 1409.14024)] and second, in the Langlands duality of Higgs bundle moduli spaces [\textit{R. Donagi} and \textit{T. Pantev}, Invent. Math. 189, No. 3, 653--735 (2012; Zbl 1263.53078)]. Both results were considered on the regular locus of the Hitchin map and it is an interesting question how they extend to the singular locus. The aim of this paper is to do the first steps in this direction. The author defines and parametrizes so-called \(\texttt{sl}(2)\)-type fibres of the \(Sp(2n,\mathbb{C})\)- and \(SO(2n+1,\mathbb{C})\)-Hitchin system. These are singular Hitchin fibres, such that spectral curve establishes a \(2\)-sheeted covering of a second Riemann surface \(Y\). This identifies the \(\texttt{sl}(2)\)-type Hitchin fibres with fibres of an \(SL(2,\mathbb{C})\)-Hitchin, respectively, \(PSL(2,\mathbb{C})\)-Hitchin map on \(Y\).
Building on results of [\textit{J. Horn}, Int. Math. Res. Not. 2022, No. 5, 3860--3917 (2022; Zbl 1482.14040)], he gives a stratification of these singular spaces by semi-abelian spectral data, studies their irreducible components and obtains a global description of the first degenerations. He also compares the semi-abelian spectral data of \(\texttt{sl}(2)\)-type Hitchin fibres for the two Langlands dual groups. This extends the well-known Langlands duality of regular Hitchin fibres to \(\texttt{sl}(2)\)-type Hitchin fibres. Finally, the author constructs solutions to the decoupled Hitchin equation for \(\texttt{sl}(2)\)-type fibres of the symplectic and odd orthogonal Hitchin system. He conjectures these to be limiting configurations along rays to the ends of the moduli space.
This paper is organized as follows: Section 1 is an introduction to the subject and summarizes the main results. In Section 2, the author introduces \(\texttt{sl}(2)\)-type Hitchin fibres of the symplectic Hitchin system. He proves the identification of these Hitchin fibres with \(\textsc{SL}(2,\mathbb{C})\)-Hitchin fibres and gives the parametrization by semi-abelian spectral data using the results of [loc. cit.]. In Section 3, he repeats these considerations for the odd orthogonal group. In Section 4, the author formulates the Langlands correspondence for \(Sp(2n,\mathbb{C})\)- and \(SO(2n+1,\mathbb{C})\)-Hitchin fibres of \(\texttt{sl}(2)\)-type. Finally, in Section 5, the author shows how to use semi-abelian spectral data for \(\texttt{sl}(2)\)-type Hitchin fibres to produce solutions to the decoupled Hitchin equation. He constructs solutions to the decouple Hitchin equation and motivate why he conjectures theses to be limiting configurations.
Reviewer: Ahmed Lesfari (El Jadida)Punctual characterization of the unitary flat bundle of weight one PVHS and application to families of curveshttps://zbmath.org/1521.140232023-11-13T18:48:18.785376Z"González-Alonso, Víctor"https://zbmath.org/authors/?q=ai:gonzalez-alonso.victor"Torelli, Sara"https://zbmath.org/authors/?q=ai:torelli.saraConsider a polarized variation of Hodge structures (PVHS) of weight one and rank \(2g\) on a smooth complex manifold \(B\), which in particular consists of a short exact sequence of holomorphic vector bundles on \(B\)
\[0\longrightarrow E=E^{1,0}\longrightarrow \mathcal{H}=\mathbb{V}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathcal{O}_B\longrightarrow E^{0,1}\longrightarrow 0\]
together with a Gauß-Manin connection on \(\mathcal{H}\). The bundle \(E\) carries a natural (maximal) flat unitary subbundle \(\mathcal{U}\subseteq E\) which encodes many properties of the natural modular map \(B\rightarrow\mathcal{A}_g\) and its relation to the (open) Torelli locus \(\mathcal{T}_g\subseteq\mathcal{A}_g\) of jacobian varieties.
The paper considers the problem of pointwise determining the fibres of the flat unitary subbundle of a PVHS of weight one. Starting from the associated Higgs field, and assuming the base has dimension \(1\), we construct a family of (smooth but possibly non-holomorphic) morphisms of vector bundles with the property that the intersection of their kernels at a general point is the fibre of the flat subbundle. We explore the first one of these morphisms in the case of a geometric PVHS arising from a family of smooth projective curves, showing that it acts as the cup-product with some sort of ``second-order Kodaira-Spencer class'' which we introduce, and check in the case of a family of smooth plane curves that this additional condition is non-trivial. The main results of the paper are the following two theorems.
\begin{itemize}
\item[1.] Let \(\dim B=1\). Then there are smooth morphisms of vector bundles
\[\eta^{\left(1\right)},\eta^{\left(2\right)},\ldots,\eta^{\left(g\right)}\colon E\rightarrow E^{0,1}\]
such that for any \(\alpha\in\Gamma\left(E\right)\) it holds
\[
\alpha\in\Gamma\left(\mathcal{U}\right)\Longleftrightarrow\eta^{\left(1\right)}\left(\alpha\right)=\eta^{\left(2\right)}\left(\alpha\right)=\ldots=\eta^{\left(g\right)}\left(\alpha\right)=0.
\]
In particular we have
\[
\mathcal{U}_b\subseteq\bigcap_{k=1}^{g}\ker\eta^{\left(k\right)}_b\subseteq E_b
\]
with equality for \(b\) in a dense Zariski-open subset of \(B\).
\item[2.] Let \(f\colon\mathcal{C}\rightarrow B\) be a family of smooth projective curves \(C_b=f^{-1}\left(b\right)\) with \(\dim B=1\). For any \(b\in B\) let \(\mathcal{K}_b\subseteq E_b=H^0\left(\omega_{C_b}\right)\) be the fibre of \(\mathcal{K}\) on \(B\), and \(\mu_b\in H^1\left(T_{C_b}\right)\) the second-order Kodaira-Spencer class of \(C_b\subseteq\mathcal{C}\). Let
\[\hat{\mu_b}\colon H^0\left(\omega_{C_b}\right)\stackrel{\mu_b\cdot}{\longrightarrow}H^1\left(\mathcal{O}_{C_b}\right)=E_b^{\vee}\twoheadrightarrow\mathcal{K}_b^{\vee}.\]
Then \(\mathcal{U}_b\subseteq\mathcal{K}_b\cap\ker\hat{\mu_b}\).
\end{itemize}
The additional morphisms of vector bundles are constructed from the Higgs field \(\theta\) by applying the connection \(\nabla^{T}\) induced on \(\mathrm{Hom}^s\left(E,E^{0,1}\right)\) by the Gauss-Manin connection. If the base \(B\) of the family is actually a submanifold of \(\mathcal{A}_g\) (or the Siegel upper-half space \(\mathbb{H}_g\)), this connection \(\nabla^{T}\) is the restriction to \(B\) of the Levi-Civita connection of the Siegel metric. Thus our construction hints again at a connection between the equality \(\mathcal{U}=\mathcal{K}\) and the existence of geodesics inside \(B\) with respect to the Siegel metric.
Reviewer: Mohammad Reza Rahmati (León)Applications of the algebraic geometry of the Putman-Wieland conjecturehttps://zbmath.org/1521.140242023-11-13T18:48:18.785376Z"Landesman, Aaron"https://zbmath.org/authors/?q=ai:landesman.aaron"Litt, Daniel"https://zbmath.org/authors/?q=ai:litt.danielLet \(\Sigma_{g,n}\) denote an orientable topological surface of genus \(g\) with \(n\) punctures. Let \(H\) be a finite group. Given a finite unramified \(H\)-cover of topological surfaces \(\Sigma_{g',n'} \to \Sigma_{g,n}\), there is an action of a finite index subgroup \(\Gamma\) of the mapping class group \(\mathrm{Mod}_{g,n+1}\) of \(\Sigma_{g,n+1}\) on
\(H_1(\Sigma_{g'}, \mathbb C)\), as we now explain. The group \(\mathrm{Mod}_{g,n+1}\) acts on \(\pi_1(\Sigma_{g,n},x)\) for some basepoint \(x\), and we can take \(\Gamma\) to be the stabilizer of the surjection \(\phi: \pi_1(\Sigma_{g,n},x) \twoheadrightarrow H\), where
\(\phi\) corresponds to the cover \(\Sigma_{g',n'} \to \Sigma_{g,n}\). Then, for \(x' \in \Sigma_{g',n'}\) mapping to \(x\), \(\Gamma\) acts on \(\pi_1(\Sigma_{g',n'},x')=\ker \phi\), preserving the conjugacy classes of the loops around the punctures in \(\Sigma_{g', n'}\), and hence acts on \(H_1(\Sigma_{g'},\mathbb C)\).
Conjecture 0.1. [\textit{A. Putman} and \textit{B. Wieland}, J. Lond. Math. Soc., II. Ser. 88, No. 1, 79--96 (2013; Zbl 1279.57014), Conjecture 1.2] Fix \(g \geq 2\) and \(n \geq 0\). For any unramified cover \(\Sigma_{g',n'} \to \Sigma_{g,n}\), the vector space \(H_1(\Sigma_{g'}, \mathbb C)\) has no nonzero vectors with finite orbit under the action of \(\Gamma\).
We now formally define what it means for \(f: X \to Y\) to furnish a counterexample to Putman-Wieland.
Definition 0.2. Suppose \(h: \Sigma_{g',n'} \to \Sigma_{g,n}\) is a finite covering of topological surfaces. Let \(\Gamma \subset \mathrm{Mod}_{g,n+1}\) denote the finite index subgroup preserving \(h\). The action of \(\Gamma\) on \(\pi_1(\Sigma_{g',n'}, \mathbb C)\) induces an action on \(H_1(\Sigma_{g',n'}, \mathbb C)\) which preserves the subspace spanned by homology classes of loops around punctures, and hence also induces an action on
\(H_1(\Sigma_{g'}, \mathbb C) \simeq H^1(\Sigma_{g'}, \mathbb C)\), via Poincaré duality.
We say \(h\) \textit{furnishes a counterexample to Putman-Wieland}, if there is some nonzero \(v \in H^1(\Sigma_{g'}, \mathbb C)\) with finite orbit under \(\Gamma\).
Let \(Y\) be a compact Riemann surface of genus \(g\) with \(n\) marked points \(p_1,
\ldots, p_n\). Upon identifying \(\pi_1(Y-\{p_1, \ldots, p_n\}) \simeq \pi_1(\Sigma_{g,n})\),
let \(f: X \to Y\) be the covering of compact Riemann surfaces, ramified at most over \(p_1, \ldots, p_n\), corresponding to the topological cover \(h\). If \(h\) furnishes a counterexample to Putman-Wieland, we also say \(f: X \to Y\) furnishes a counterexample to Putman-Wieland.
Suppose \(h\) is Galois with Galois group \(H\), and \(\rho\) is an irreducible \(H\)-representation. Note that \(H\) and \(\Gamma\) simultaneously act on \(H^1(\Sigma_{g'}, \mathbb C)\). We say a counterexample to Putman-Wieland is \textit{\(\rho\)-isotypic}
if every element of the the \(\rho\)-isotypic subspace \(H^1(\Sigma_{g'}, \mathbb C)^\rho\) has finite orbit under \(\Gamma\). Here if \(V\) is an \(H\)-representation, \(V^\rho\) denotes the \(\rho\)-isotypic subspace of \(V\), i.e.~the image of the natural evaluation map
\[\mathrm{Hom}_H(\rho, V)\otimes \rho\to V.\]
Remark 0.3. Even though the Putman-Wieland conjecture assumes \(g \geq 2\), we still say \(f: X
\to Y\) furnishes a counterexample to Putman-Wieland when \(Y\) has genus \(g \leq 1\).
Here is the slight improvement on [\textit{V. Marković} and \textit{O. Tošić}, The second variation of the Hodge norm and higher Prym representations. \url{https://people.maths.ox.ac.uk/~markovic/MT-hodge.pdf}, Theorem 1.5]. We let \(\lambda_1(X)\) denote the smallest nonzero eigenvalue of the Laplacian acting on \(L^2\) functions on \(X\).
Theorem 0.4. Suppose \(Y\) has genus \(g \geq 2\) and \(f: X \to Y\) furnishes a counterexample to Putman-Wieland. Then \(\frac{1}{g-1} \geq 2 \lambda_1(X)\). If \(f\) is Galois, then
\(\frac{1}{g-1} > 2 \lambda_1(X)\).
We next prove a result ruling out \(\rho\)-isotypic counterexamples to Putman-Wieland in genus at least \(2\).
Theorem 0.5. Suppose \(f: X\to Y\) is a Galois \(H\)-cover furnishing a counterexample to
Putman-Wieland which is \(\rho\)-isotypic in the sense of Definition 0.2.
Then \(Y\) has genus at most \(1\).
Equivalently, if the genus of \(Y\) is at least \(2\), for each irreducible \(H\)-representation \(\rho\), there exists an element of \(H^1(X, \mathbb{C})^\rho\) with infinite orbit under the virtual action of the mapping class group of \(Y\).
This will follow from a stronger statement:
Theorem 0.6. Let \(X\to Y\) be a Galois \(H\)-cover, where \(Y\) has genus at least \(2\). Let \(\rho\) be an irreducible complex \(H\)-representation. Then the virtual action of the mapping class group of \(Y\) on \(H^1(X, \mathbb{C})^\rho\) is not unitary.
Theorem 0.5 immediately implies the following.
Corollary 0.7. Let \(f' : X' \to Y\) be the Galois closure of the counterexample to Putman-Wieland \(f: X \to Y\) from [\textit{V. Marković}, Bull. Lond. Math. Soc. 54, No. 6, 2324--2337 (2022; Zbl 07740276)]. For any irreducible \(H\) representation \(\rho\), \(f'\) is not a \(\rho\)-isotypic counterexample to Putman-Wieland.
Reviewer: Mohammad Reza Rahmati (León)On the intersection cohomology of the moduli of \(\mathrm{SL}_n\)-Higgs bundles on a curvehttps://zbmath.org/1521.140252023-11-13T18:48:18.785376Z"Maulik, Davesh"https://zbmath.org/authors/?q=ai:maulik.davesh"Shen, Junliang"https://zbmath.org/authors/?q=ai:shen.junliangThe aim of this paper is to explore cohomological structures for the moduli space of degree \(d\) semistable \(SL_n\)-Higgs bundles on a nonsingular irreducible projective curve of genus \(g\geq2\), with respect to an effective divisor of degree \(\mbox{deg}>2g-2\). The authors show that the support theorem [\textit{M. A. de Cataldo}, Compos. Math. 153, No. 6, 1316--1347 (2017; Zbl 1453.14029)] and the topological mirror symmetry conjecture [\textit{M. Groechenig} et al., Invent. Math. 221, No. 2, 505--596 (2020; Zbl 1451.14123); \textit{T. Hausel} and \textit{M. Thaddeus}, Invent. Math. 153, No. 1, 197--229 (2003; Zbl 1043.14011); \textit{D. Maulik} and \textit{J. Shen}, Forum Math. Pi 9, Paper No. e8, 49 p. (2021; Zbl 1478.14054)], which were proven in the case \(gcd(n,d)=1\), actually hold for arbitrary \(d\).
More precisely, the authors explore the cohomological structure for the possibly singular moduli of \(SL_n\)-Higgs bundles for arbitrary degree on a genus \(g\) curve with respect to an effective divisor of degree \(>2g-2\). They prove a support theorem for the \(SL_n\)-Hitchin fibration extending de Cataldo's support theorem in the nonsingular case, and a version of the Hausel-Thaddeus topological mirror symmetry conjecture for intersection cohomology. As an immediate application of these results, the authors also give a proof of a generalized version of the Harder-Narasimhan theorem [\textit{G. Harder} and \textit{M. S. Narasimhan}, Math. Ann. 212, 215--248 (1975; Zbl 0324.14006)] for intersection cohomology and arbitrary degree. The main tool is an Ngô-type support inequality established recently in [\textit{D. Maulik} and \textit{J. Shen}, Geom. Topol. 27, No. 4, 1539--1586 (2023; Zbl 1514.14038)] which works for possibly singular ambient spaces and intersection cohomology complexes.
Reviewer: Ahmed Lesfari (El Jadida)Stable bundles of rank 2 with Chern's classes \(c_1=0\), \(c_2=2\) on \(\mathbb{P}^3\) and Poncelet hyperquadricshttps://zbmath.org/1521.140262023-11-13T18:48:18.785376Z"Tikhomirov, Sergeĭ A."https://zbmath.org/authors/?q=ai:tikhomirov.sergey-aSummary: In this article we investigate the variety \(M(0,2)\) of stable vector bundles of rank 2 on \(\mathbb{P}^3\) with Chern's classes \(c_1=0\), \(c_2=2\) and give the explicit description of closure of \(M(0,2)\) as the intersection of special determinantal locus with uniquely determined Poncelet hyperquadric in \(\mathbb{P}^{20}\).Indeterminacy loci of iterate maps in moduli spacehttps://zbmath.org/1521.140272023-11-13T18:48:18.785376Z"Kiwi, Jan"https://zbmath.org/authors/?q=ai:kiwi.jan"Nie, Hongming"https://zbmath.org/authors/?q=ai:nie.hongmingThe space of rational maps \(f:\mathbb{P}^1\to \mathbb{P}^1\) of degree \(d\), denoted by \(\mathrm{Rat}_d\), admits a compactification \(\overline{\mathrm{Rat}_d}\) and a natural morphism \(\Psi_n:\mathrm{Rat}_d\to\mathrm{Rat}_{d^n}\) defined by iteration: \([f]\mapsto [f^n]\). This morphism extends to a rational map \(\Psi_n:\overline{\mathrm{Rat}_d}\dashrightarrow \overline{\mathrm{Rat}_{d^n}}\) whose indeterminacy locus \(I(d)\) is understood from the work of \textit{L. DeMarco} [Duke Math. J. 130, No. 1, 169--197 (2005; Zbl 1183.37086)] and is independent of \(n\). In this work the focus is on the moduli space of rational maps \(\mathrm{rat}_d\), that is the quotient of \(\mathrm{Rat}_d\) by the action of the group \(PSL(2)\). Here we have a GIT compactification \(\overline{\mathrm{rat}_d}\) and a morphism defined by iteration \(\Phi_n:\mathrm{rat}_d\to\mathrm{rat}_{d^n}\) which extends to a rational map \(\Phi_n:\overline{\mathrm{rat}_d}\to \overline{\mathrm{rat}_{d^n}}\). The indeterminacy locus of \(\Phi_n\) is described as the union of the image of \(I(d)\) by the quotient map and the set \(\mathcal{U}_n\) of maps whose \(n\)-iteration is strictly semistable with respect to the action of \(PSL(2)\). The set \(\mathcal{U}_n\) is described as an algebraic set that is increasing in \(n\), in particular they find \(\mathcal{U}_n\subsetneq \mathcal{U}_{n+1}\) for even degree \(d\), and \(\mathcal{U}_n=\mathcal{U}_5\) for \(d=3\), \(n\geq 5\). These characterizations are obtained by the interplay between strictily semi-stable points \([f]\) and \(1\)-parameter families of rational maps \(f_t\) via Hilbert-Mumford criterion, and more notably by the interplay between families \(f_t\) of rational maps and rational maps \(f_t:\mathbf{P}^1\to \mathbf{P}^1\) defined in the \textit{Berkovich projective line} \(\mathbf{P}^1\), regarding this as a Dynamical system on the connected compact Haussdorf space \(\mathbf{P}^1\). The interesting dynamical properties of \([f_t]\) are then related to the behavior of \(\Phi_n\) on the point \([f]\) with \([f]=\lim_{t\to 0} [f_t]\).
Reviewer: Federico Quallbrunn (Buenos Aires)``Scheme-theoretic images'' of morphisms of stackshttps://zbmath.org/1521.140282023-11-13T18:48:18.785376Z"Emerton, Matthew"https://zbmath.org/authors/?q=ai:emerton.matthew"Gee, Toby"https://zbmath.org/authors/?q=ai:gee.tobySummary: We give criteria for certain morphisms from an algebraic stack to a (not necessarily algebraic) stack to admit an (appropriately defined) scheme-theoretic image. We apply our criteria to show that certain natural moduli stacks of local Galois representations are algebraic (or Ind-algebraic) stacks.An embedding problem for finite local torsors over twisted curveshttps://zbmath.org/1521.140292023-11-13T18:48:18.785376Z"Otabe, Shusuke"https://zbmath.org/authors/?q=ai:otabe.shusukeSummary: In his previous paper, the author proposed as a problem a purely inseparable analogue of the Abhyankar conjecture for affine curves in positive characteristic and gave a partial answer to it, which includes a complete answer for finite local nilpotent group schemes. In the present paper, motivated by the Abhyankar conjectures with restricted ramifications due to Harbater and Pop, we study a refined version of the analogous problem, based on a recent work on tamely ramified torsors due to Biswas-Borne, which is formulated in terms of root stacks. We study an embedding problem to conclude that the refined analogue is true in the solvable case.
{{\copyright} 2021 Wiley-VCH GmbH}Constant scalar curvature Kähler metrics on rational surfaceshttps://zbmath.org/1521.140302023-11-13T18:48:18.785376Z"Martinez-Garcia, Jesus"https://zbmath.org/authors/?q=ai:martinez-garcia.jesusSummary: We consider projective rational strong Calabi dream surfaces: projective smooth rational surfaces which admit a constant scalar curvature Kähler metric for every Kähler class. We show that there are only two such rational surfaces, namely the projective plane and the quadric surface. In particular, we show that all rational surfaces other than those two admit a destabilising slope test configuration for some polarisation, as introduced by Ross and Thomas. We further show that all Hirzebruch surfaces other than the quadric surface and all rational surfaces with Picard rank 3 do not admit a constant scalar curvature Kähler metric in any Kähler class.
{{\copyright} 2021 Wiley-VCH GmbH}Jordan constant for the Cremona group of rank two over a finite fieldhttps://zbmath.org/1521.140312023-11-13T18:48:18.785376Z"Vikulova, A. V."https://zbmath.org/authors/?q=ai:vikulova.anastasia-v(no abstract)Trisecant flops, their associated \(K3\) surfaces and the rationality of some cubic fourfoldshttps://zbmath.org/1521.140322023-11-13T18:48:18.785376Z"Russo, Francesco"https://zbmath.org/authors/?q=ai:russo.francesco.1"Staglianò, Giovanni"https://zbmath.org/authors/?q=ai:stagliano.giovanniA very general cubic fourfold is conjectured to be irrational and the locus of rational ones, also conjecturally, should be the union of certain irreducible divisors \(\mathcal{C}_d\) (in their moduli \(\mathcal{C}\)), of special admissible cubic fourfolds of discriminant \(d\); their rationality relies on the existence of certain \(K3\) surface (further references on this \textit{ Kuznetsov Conjecture} can be found in the Introduction of the paper under review). In this paper, the authors take the point of view of Mori Theory to describe the cases \(d=14, 26, 38\) and \(42\), in fact the first four admissible values of \(d\) (cubics fourfolds in \(\mathcal{C}_d\), \(d=14,26,38\), were known to be rational); and furthermore to prove (see Theorem 5.12) the rationality of every cubic fourfold in \(\mathcal{C}_{42}\), the first not known case. The birational maps from \(X\) to a rational fourfold \(W\) are displayed in a, say, Mori Theory diagram (see (0.1)) in such a way that the role of the \(K3\) surface is very explicit: a non-minimal birational model in \(W\) can be constructed via some very peculiar linear systems of hyperplane sections.
Reviewer: Roberto Muñoz (Madrid)Desingularization of Kontsevich's compactification of twisted cubics in \(V_5\)https://zbmath.org/1521.140332023-11-13T18:48:18.785376Z"Chung, Kiryong"https://zbmath.org/authors/?q=ai:chung.kiryongSummary: By definition, the del Pezzo 3-fold \(V_5\) is the intersection of \(\text{Gr}(2,5)\) with three hyperplanes in \(\mathbb{P}^9\) under Plücker embedding, and rational curves in \(V_5\) have been examined in various studies on Fano geometry. In this paper, we propose an explicit birational relation for the Kontsevich and Simpson compactifications of twisted cubic curves in \(V_5\). As a direct corollary, we obtain a desingularized model of Kontsevich compactification that induces the intersection cohomology group of Kontsevich's space.Bioriented flags and resolutions of Schubert varietieshttps://zbmath.org/1521.140342023-11-13T18:48:18.785376Z"Cibotaru, Daniel"https://zbmath.org/authors/?q=ai:cibotaru.daniel-fSummary: We use incidence relations running in two directions in order to construct a Kempf-Laksov type resolution for any Schubert variety of the complete flag manifold but also an embedded resolution for any Schubert variety in the Grassmannian. These constructions are alternatives to the celebrated Bott-Samelson resolutions. The second process led to the introduction of \(W\)-flag varieties, algebro-geometric objects that interpolate between the standard flag manifolds and products of Grassmannians, but which are singular in general. The surprising simple desingularization of a particular such type of variety produces an embedded resolution of the Schubert variety within the Grassmannian.On the geometry of elliptic pairshttps://zbmath.org/1521.140352023-11-13T18:48:18.785376Z"Pratt, Elizabeth"https://zbmath.org/authors/?q=ai:pratt.elizabethSummary: An \textit{elliptic pair} \((X, C)\) is a projective rational surface \(X\) with \(\log\) terminal singularities, and an irreducible curve \(C\) contained in the smooth locus of \(X\), with arithmetic genus 1 and self-intersection 0. They are a useful tool for determining whether the pseudo-effective cone of \(X\) is polyhedral [\textit{A.-M. Castravet} et al., ``Blown-up toric surfaces with non-polyhedral effective cone'', Preprint, \url{arXiv:2009.14298}], and interesting algebraic and geometric objects in their own right. Especially of interest are toric elliptic pairs, where \(X\) is the blow-up of a projective toric surface at the identity element of the torus. In this paper, we classify all toric elliptic pairs of Picard number two. Strikingly, it turns out that there are only three of these. Furthermore, we study a class of non-toric elliptic pairs coming from the blow-up of \(\mathbb{P}^2\) at nine points on a nodal cubic, in characteristic \(p\). This construction gives us examples of surfaces where the pseudo-effective cone is non-polyhedral for a set of primes \(p\) of positive density, and, assuming the generalized Riemann hypothesis, polyhedral for a set of primes \(p\) of positive density.Classification and syzygies of smooth projective varieties with 2-regular structure sheafhttps://zbmath.org/1521.140362023-11-13T18:48:18.785376Z"Kwak, Sijong"https://zbmath.org/authors/?q=ai:kwak.sijong"Park, Jinhyung"https://zbmath.org/authors/?q=ai:park.jinhyungSummary: The geometric and algebraic properties of smooth projective varieties with 1-regular structure sheaf are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: smooth projective varieties with 2-regular structure sheaf. First, we give a classification of such varieties using adjunction mappings. Next, under suitable conditions, we study the syzygies of section rings of those varieties to understand the structure of the Betti tables, and show a sharp bound for Castelnuovo-Mumford regularity.Derived categories of centrally-symmetric smooth toric Fano varietieshttps://zbmath.org/1521.140372023-11-13T18:48:18.785376Z"Ballard, Matthew R."https://zbmath.org/authors/?q=ai:ballard.matthew-robert"Duncan, Alexander"https://zbmath.org/authors/?q=ai:duncan.alexander"McFaddin, Patrick K."https://zbmath.org/authors/?q=ai:mcfaddin.patrick-kSummary: We exhibit full exceptional collections of vector bundles on any smooth, Fano arithmetic toric variety whose split fan is centrally symmetric
{{\copyright} 2022 Wiley-VCH GmbH}Some ways to reconstruct a sheaf from its tautological image on a Hilbert scheme of pointshttps://zbmath.org/1521.140382023-11-13T18:48:18.785376Z"Krug, Andreas"https://zbmath.org/authors/?q=ai:krug.andreas.1|krug.andreas"Rennemo, Jørgen Vold"https://zbmath.org/authors/?q=ai:rennemo.jorgen-voldSummary: For \(X\) a smooth quasi-projective variety and \(X^{[n]}\) its associated Hilbert scheme of \(n\) points, we study two canonical Fourier-Mukai transforms \({\mathsf{D}}(X) \to {\mathsf{D}} \big(X^{[n]}\big)\), the one along the structure sheaf and the one along the ideal sheaf of the universal family. For \(\operatorname{\mathsf{dim}} X \geq 2\), we prove that both functors admit a left inverse. This means in particular that both functors are faithful and injective on isomorphism classes of objects. Using another method, we also show in the case of an elliptic curve that the Fourier-Mukai transform along the structure sheaf of the universal family is faithful and injective on isomorphism classes. Furthermore, we prove that the universal family of \(X^{[n]}\) is always flat over \(X\), which implies that the Fourier-Mukai transform along its structure sheaf maps coherent sheaves to coherent sheaves.
{{\copyright} 2021 The Authors. Mathematische Nachrichten published by Wiley-VCH GmbH}Vanishing for Hodge ideals on toric varietieshttps://zbmath.org/1521.140392023-11-13T18:48:18.785376Z"Dutta, Yajnaseni"https://zbmath.org/authors/?q=ai:dutta.yajnaseniSummary: In this article we construct a Koszul-type resolution of the \(p^{\text{th}}\) exterior power of the sheaf of holomorphic differential forms on smooth toric varieties and use this to prove a Nadel-type vanishing theorem for Hodge ideals associated to effective \(\mathbb{Q}\)-divisors on smooth projective toric varieties. This extends earlier results of Mustaţă and Popa.Tilting correspondences of perfectoid ringshttps://zbmath.org/1521.140402023-11-13T18:48:18.785376Z"Kundu, Arnab"https://zbmath.org/authors/?q=ai:kundu.arnabSummary: In this article, we present an alternate proof of a vanishing result of étale cohomology on perfectoid rings due to Česnavičius and more recently proved by a different approach by \textit{B. Bhatt} and \textit{P. Scholze} [Ann. Math. (2) 196, No. 3, 1135--1275 (2022; Zbl 07611906)]. To establish that, we prove a tilting equivalence of étale cohomology of perfectoid rings taking values in commutative, finite étale group schemes. On the way, we algebraically establish an analogue of the tilting correspondences of \textit{P. Scholze} [Publ. Math., Inst. Hautes Étud. Sci. 116, 245--313 (2012; Zbl 1263.14022)], between the category of finite étale schemes over a perfectoid ring and that over its tilt, without using tools from almost ring theory or adic spaces.\(\mathbb{A}^1\)-connected components of blowup of threefolds fibered over a surfacehttps://zbmath.org/1521.140412023-11-13T18:48:18.785376Z"Pawar, Rakesh"https://zbmath.org/authors/?q=ai:pawar.rakesh-rSummary: Over a perfect field, we determine the sheaf of \(\mathbb{A}^1\)-connected components of a class of threefolds given by the blowup of a variety admitting a \(\mathbb{P}^1\)-fibration over an \(\mathbb{A}^1\)-rigid surface, along a smooth curve. Over an infinite perfect field \(k\), we determine the sheaf of \(\mathbb{A}^1\)-connected components of a class of threefolds given by the blowup of a variety admitting a \(\mathbb{P}^1\)-fibration over a non-uniruled surface. As a consequence, in the respective cases, we verify that the sheaf of \(\mathbb{A}^1\)-connected components for such varieties is \(\mathbb{A}^1\)-invariant.Exceptional loci in Lefschetz theoryhttps://zbmath.org/1521.140422023-11-13T18:48:18.785376Z"Raskin, Sam"https://zbmath.org/authors/?q=ai:raskin.sam"Smith, Geoffrey"https://zbmath.org/authors/?q=ai:smith.geoffrey-s|smith.geoffrey|smith.geoffrey-howard|smith.geoffrey-l|smith.geoffrey-d|smith.geoffrey-bSummary: Let \(\phi :X\rightarrow \mathbb{P}^n\) be a morphism of varieties. Given a hyperplane \(H\) in \(\mathbb{P}^n\), there is a Gysin map from the compactly supported cohomology of \(\phi^{-1}(H)\) to that of \(X\). We give conditions on the degree of the cohomology under which this map is an isomorphism for all but a low-dimensional set of hyperplanes, generalizing results due to \textit{A. N. Skorobogatov} [Isr. J. Math. 80, No. 3, 359--379 (1992; Zbl 0774.11045)], \textit{O. Benoist} [Bull. Soc. Math. Fr. 139, No. 4, 555--569 (2011; Zbl 1244.14045)], and \textit{B. Poonen} and \textit{K. Slavov} [Int. Math. Res. Not. 2022, No. 6, 4503--4513 (2022; Zbl 1483.14097)]. Our argument is based on Beilinson's theory of singular supports for étale sheaves.
{{\copyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}Comparison of Kummer logarithmic topologies with classical topologieshttps://zbmath.org/1521.140432023-11-13T18:48:18.785376Z"Zhao, Heer"https://zbmath.org/authors/?q=ai:zhao.heerSummary: We compare the Kummer flat (resp., Kummer étale) cohomology with the flat (resp., étale) cohomology with coefficients in smooth commutative group schemes, finite flat group schemes, and Kato's logarithmic multiplicative group. We are particularly interested in the case of algebraic tori in the Kummer flat topology. We also make some computations for certain special cases of the base log scheme.The first and second homotopy groups of a homogeneous space of a complex linear algebraic grouphttps://zbmath.org/1521.140442023-11-13T18:48:18.785376Z"Borovoi, Mikhail"https://zbmath.org/authors/?q=ai:borovoi.mikhail-vThe author considers a connected linear algebraic group \(G\) over the complex numbers. If a subgroup \(H\) is connected, the topological fundamental group of the homogeneous space \(G/H\) is computed algebraically: it is canonically isomorphic to the cokernel of the map between the respective algebraic fundamental groups of \(H\) and \(G\), induced by inclusion.
Two different proofs are given, the first one assuming that the Picard group of \(G\) is trivial, the second one in the general case.
The second homotopy group of \(G/H\) is also computed algebraically, without connectedness assumption on \(H\).
Reviewer: Matilde Maccan (Rennes)The fundamental group of the complement of the Hesse configurationhttps://zbmath.org/1521.140452023-11-13T18:48:18.785376Z"Kaneko, Jyoichi"https://zbmath.org/authors/?q=ai:kaneko.jyoichiSummary: This note is of supplementary nature to our previous paper [\textit{J. Kaneko} et al., Int. J. Math. 31, No. 3, Article ID 2050021, 27 p. (2020; Zbl 1439.33005)]. Let \(S\) be the union of the nodal cubic curve and its three inflectional tangents in the complex projective plane \(\mathbb{P}^2\). Such \(S\) has appeared as the singular locus of certain hypergeometric system introduced in [\textit{J. Kaneko} et al., Int. J. Math. 28, No. 3, Article ID 1750015, 34 p. (2017; Zbl 1364.33016)], and we have given generators and defining relations of the fundamental group of \(X= \mathbb{P}^2\smallsetminus S\) [\textit{J. Kaneko} et al., Int. J. Math. 31, No. 3, Article ID 2050021, 27 p. (2020; Zbl 1439.33005)]. \(X\) has a 9-fold Galois covering space \(\widetilde{X}\) given by the complement of the Hesse configuration of 12 lines in \(\mathbb{P}^2\). Hence one can apply the method of Reidemeister-Schreier to derive a finite presentation of \(\pi_1(\widetilde{X})\), which we carry out in this note.Quadratic Chabauty and \(p\)-adic Gross-Zagierhttps://zbmath.org/1521.140462023-11-13T18:48:18.785376Z"Hashimoto, Sachi"https://zbmath.org/authors/?q=ai:hashimoto.sachiSummary: Let \(X\) be a quotient of the modular curve \(X_0(N)\) whose Jacobian \(J_X\) is a simple factor of \(J_0(N)^{new}\) over \(\mathbf{Q} \). Let \(f\) be the newform of level \(N\) and weight 2 associated with \(J_X\); assume \(f\) has analytic rank 1. We give analytic methods for determining the rational points of \(X\) using quadratic Chabauty by computing two \(p\)-adic Gross-Zagier formulas for \(f\). Quadratic Chabauty requires a supply of rational points on the curve or its Jacobian; this new method eliminates this requirement. To achieve this, we give an algorithm to compute the special value of the anticyclotomic \(p\)-adic \(L\)-function of \(f\) constructed by \textit{M. Bertolini} et al. [Duke Math. J. 162, No. 6, 1033--1148 (2013; Zbl 1302.11043)], which lies outside of the range of interpolation.Urata's theorem in the logarithmic case and applications to integral pointshttps://zbmath.org/1521.140472023-11-13T18:48:18.785376Z"Javanpeykar, Ariyan"https://zbmath.org/authors/?q=ai:javanpeykar.ariyan"Levin, Aaron"https://zbmath.org/authors/?q=ai:levin.aaronSummary: \textit{T. Urata} [Tôhoku Math. J. (2) 31, 349--353 (1979; Zbl 0404.32013)] showed that a pointed compact hyperbolic variety admits only finitely many maps from a pointed curve. We extend Urata's theorem to the setting of (not necessarily compact) hyperbolically embeddable varieties. As an application, we show that a hyperbolically embeddable variety over a number field \(K\) with only finitely many \(\mathcal{O}_{L,T} \)-points for any number field \(L/K\) and any finite set of finite places \(T\) of \(L\) has, in fact, only finitely many points in any given \(\mathbb{Z} \)-finitely generated integral domain of characteristic zero. We use this latter result in combination with Green's criterion for hyperbolic embeddability to obtain novel finiteness results for integral points on symmetric self-products of smooth affine curves and on complements of large divisors in projective varieties. Finally, we use a partial converse to Green's criterion to further study hyperbolic embeddability (or its failure) in the case of symmetric self-products of curves. As a by-product of our results, we obtain the first example of a smooth affine Brody-hyperbolic threefold over \(\mathbb{C}\) which is not hyperbolically embeddable.
{{\copyright} 2022 The Authors. \textit{Bulletin of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.}Rational points on certain homogeneous varietieshttps://zbmath.org/1521.140482023-11-13T18:48:18.785376Z"Yang, Pengyu"https://zbmath.org/authors/?q=ai:yang.pengyuSummary: Let \(L\) be a simply-connected simple connected algebraic group over a number field \(F\), and \(H\) be a semisimple absolutely maximal connected \(F\)-subgroup of \(L\). Let \(\Delta (H)\) be the image of \(H\) diagonally embedded in \(L^n\). Under a cohomological condition, we prove an asymptotic formula for the number of rational points of bounded height on projective equivariant compactifications of \(\Delta (H)\backslash L^n\) with respect to a balanced line bundle.Enumerating odd-degree hyperelliptic curves and abelian surfaces over \(\mathbb{P}^1\)https://zbmath.org/1521.140492023-11-13T18:48:18.785376Z"Han, Changho"https://zbmath.org/authors/?q=ai:han.changho"Park, Jun-Yong"https://zbmath.org/authors/?q=ai:park.jun-yongSummary: Given asymptotic counts in number theory, a question of Venkatesh asks what is the topological nature of lower order terms. We consider the arithmetic aspect of the inertia stack of an algebraic stack over finite fields to partially answer this question. Subsequently, we acquire new sharp enumerations of quasi-admissible odd-degree hyperelliptic curves over \(\mathbb{F}_q(t)\) ordered by discriminant height.On orthogonal local models of Hodge typehttps://zbmath.org/1521.140502023-11-13T18:48:18.785376Z"Zachos, Ioannis"https://zbmath.org/authors/?q=ai:zachos.ioannisSummary: We study local models that describe the singularities of Shimura varieties of non-PEL type for orthogonal groups at primes where the level subgroup is given by the stabilizer of a single lattice. In particular, we use the Pappas-Zhu construction and we give explicit equations that describe an open subset around the ``worst'' point of orthogonal local models given by a single lattice. These equations display the affine chart of the local model as a hypersurface in a determinantal scheme. Using this, we prove that the special fiber of the local model is reduced and Cohen-Macaulay.Meromorphic functions without real critical values and related braidshttps://zbmath.org/1521.140512023-11-13T18:48:18.785376Z"Libgober, Anatoly"https://zbmath.org/authors/?q=ai:libgober.anatoly-s"Shapiro, Boris"https://zbmath.org/authors/?q=ai:shapiro.boris-zalmanovich|shapiro.borisLet \(\mathcal H_{d,g}\) denote the Hurwitz space of degree \(d\) meromorphic functions on genus Riemann surfaces. Motivated by the phenomenon of avoided level crossings of perturbations from mathematical physics, the authors study the open subset \(\mathcal H^{nr}_{g,d} \subset \mathcal H_{d,g}\) consisting of meromorphic functions with simple critical values, none of which is real. If \(f:E \to \mathbb C \mathbb P^1\) is a meromorphic function from \(\mathcal H^{nr}_{g,d}\), the pre-image \(N_f = f^{-1}(\mathbb R \mathbb P^1) \subset E\) is a disjoint union of smooth closed real curves \(O_1, \dots O_l\) with orientation given by the positive direction of \(\mathbb R \subset \mathbb R \mathbb P^1\) and the connected components of \(E \setminus N_f\) are separated into those mapping onto the positive and negative half planes, so that \(N_f\) is a separating oriented collection of ovals in the authors' terminology. Each oval \(O_j\) is assigned a positive integer \(m_j\) given by the degree of the map to \(\mathbb R \mathbb P^1 \cong S^1\) so that \(\sum m_j = d\). The key idea is to assign to \(f\) the isotopy class of the map \(S^1 \to (E^d/\Delta)/\mathrm{Sym}_d\) given by \(t \mapsto f^{-1} (t) \subset E\) considered in the Braid group \(\mathrm{Br}_d (E)\) on \(d\) strands. As seen in [\textit{J. S. Birman}, Braids, links, and mapping class groups. Based on lecture notes by James Cannon. Princeton, NJ: Princeton University Press (1975; Zbl 0305.57013)], this associates connected components of \(\mathcal H^{nr}_{g,d}\) to certain equivalence classes of braids.
In the paper under review, the authors introduce subgroups of \textit{boundary braids} to which the braids of meromorphic functions naturally belong. They describe the boundary braids in terms of standard generators, viewing them in terms of configuration spaces. Furthermore, they describe the classes in the mapping class group \(\mathrm{Mod}(E)\) corresponding to the boundary braids via the Nielsen-Thurston classification [\textit{B. Farb} and \textit{D. Margalit}, A primer on mapping class groups. Princeton, NJ: Princeton University Press (2011; Zbl 1245.57002)]. Using their classification of the braids, the main theorem describes the connected components of \(\mathcal H^{nr}_{g,d}\) as the orbits of the mapping class group of a closed surface acting on the conjugacy classes of subgroups of boundary braids (when \(g=0\) or \(g=1\), the action is on the quotient of the braid group by the respective centers). As a consequence they recover several results of \textit{S. M. Natanzon} [Sov. Math., Dokl. 30, 724--726 (1984; Zbl 0599.14021); translation from Dokl. Akad. Nauk SSSR, 279, 803--805 (1984); \textit{S. M. Natanzon}, Sel. Math. 12, No. 3, 1 (1991; Zbl 0801.30034); translation from Tr. Semin. Vektorn. Tenzorn. Anal. 23, 79--103 (1988) and ibid. 24, 104--132 (1988)]. They apply their results to some special classes of meromorphic functions, such as those induced from generic projections of plane curves. The paper closes with some open problems.
Reviewer: Scott Nollet (Fort Worth)A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curveshttps://zbmath.org/1521.140522023-11-13T18:48:18.785376Z"Battistella, Luca"https://zbmath.org/authors/?q=ai:battistella.luca"Carocci, Francesca"https://zbmath.org/authors/?q=ai:carocci.francescaThe boundary components of the space of stable maps to projective space may in higher genus have excess dimension. The goal of this paper is an explicit modular desingularisation of the main component of the moduli space of stable maps to projective space in genus two and degree \(d\geq 3\). More precisely, there exists a logarithmically smooth and proper DM stack \(\mathcal{VZ}_{2,n}(\mathbb P^r,d)\) mapping to the main component. Families of curves of genus two are described by double covers of families of rational curves. Besides the curves with isolated singularities studied earlier by the first author also non-reduced curves occur.
Reviewer: Jan Stevens (Göteborg)Restricted tangent bundles for general free rational curveshttps://zbmath.org/1521.140532023-11-13T18:48:18.785376Z"Lehmann, Brian"https://zbmath.org/authors/?q=ai:lehmann.brian"Riedl, Eric"https://zbmath.org/authors/?q=ai:riedl.ericLet \(X\) be a smooth complex projective variety of dimension \(n\) which carries a family of free rational curves, so the locus \(\mathrm{Rat} (X) \subset \overline M_{0,0} (X)\) parametrizing stable maps \(f: \mathbb P^1 \to X\) with \(f^* T_X\) globally generated is non-empty. For each irreducible component \(M \subset \overline{\mathrm{Rat} (X)}\), there is a unique irreducible component \(M^\prime \subset \overline{M}_{0,1} (X)\) lying over \(M\) that comes with an evaluation map \(M^\prime \to X\). Taking an irreducible component \(M\) for which the composition of the normalization \(\tilde M^\prime \to M^\prime\) and the evaluation map \(M^\prime \to X\) has connected fibers, the authors define the slope of a torsion free sheaf \(\mathcal E\) on \(X\) by \(\mu_C (\mathcal E) = c_1 (\mathcal E) \cdot C / \mathrm{rank} \; \mathcal E\), where \(C = f_* \mathbb P^1\) is a general member of \(M\) and \(c_1 (\mathcal E)\) is the first Chern class. This definition of slope leads to two filtrations of \(\mathcal E|_C\): the restriction of the Harder-Narasimhan (HN) filtration on \(X\) to \(C\) and the HN filtration of \(\mathcal E|_C\) on \(C\). The authors show that these two filtrations are close in a sense made precise.
The first result says that if \(\mathcal E\) is stable of rank \(r\) with respect to the curve class \([C]\) and \(\mathcal E|_C \cong \oplus \mathcal O (a_i)\), then \(|\mu_C (\mathcal E) - a_i| < r/2\) for each \(i\). This follows from a recent generalization of the Grauert-Mulich thoerem due to \textit{A. Patel} et al. [``Moduli of linear slices of high degree hypersurfaces'', Preprint, \url{arXiv:2005.03689}]. Looking at the filtrants of the HN filtration \(0 = \mathcal E_0 \subset \mathcal E_1 \subset \dots \mathcal E_s = \mathcal E\) and taking \(\vec{v}\) to be the non-increasing \(r\)-tuple tuple that for \(0 < j < r\) contains \(r_j = \mathrm{rank}(\mathcal E_j/\mathcal E_{j-1})\) copies of \(\mu_C (\mathcal E_j/\mathcal E_{j-1})\), they show that each component of the vector \(\vec{v} - (a_1, \dots, a_r)\) is less than \(\sup \{r_j\} / 2\). When \(\mathcal E = T_X\) is the tangent bundle, they define the \textit{slope panel} of \(C\) by \(\mathrm{SP} (C) = (a_1, \dots, a_r)/\mu_C (T_X)\) and the \textit{expected slope panel} of \(C\) by \({\mathrm ESP} (C) = \vec{v} / \mu_C (T_X)\). For \(\epsilon > 0\) fixed, they conclude that if \(C\) has anticanonical degree \(>n^2/2 \epsilon\), then \(|\mathrm{ESP} (C) - \mathrm{SP} (C)|_{\mathrm{sup}} < \epsilon\).
The main result says that if \(r = \mathrm{rank} \; \mathcal E \leq 5\), then there is a constant \(\Gamma (r)\) such that \(\mathcal E\) is stable with respect to \([C]\) if and only if there is a curve \(\widetilde C\) obtained by gluing and smoothing \(\Gamma (r)\) copies of \(C\) with \(\mathcal E|_{\widetilde C}\) semistable. When \(\mathcal E = T_X\) is the tangent bundle and \(n \leq 5\), it follows that there is a constant \(k\) for which a general curve \(C\) specializing to a union of \(k\) rational curves intersecting at a point satisfies \(\mathrm{SP} (C) = \mathrm{ESP} (C)\).
The authors give applications when \(X\) is a Fano variety. The first says that the set of restricted tangent bundles is controlled by a finite set of data. The second involves the geometric Manin conjecture studied by \textit{B. Lehmann} and \textit{S. Tanimoto} [Compos. Math. 155, No. 5, 833--862 (2019; Zbl 1451.14089)], which predicts the behavior of discrete invariants of \(\overline{\mathrm{Rat} (X)}\) such as dimension and number of irreducible components as the degree increases. Specifically, the authors analyze the analog of the formulation Manin's conjecture for rational points due to \textit{E. Peyre} [Doc. Math. 22, 1615--1659 (2017; Zbl 1434.11084)] in the setting of rational curves.
Reviewer: Scott Nollet (Fort Worth)\(\mathbb{Z}/r \mathbb{Z}\)-equivariant covers of \(\mathbb{P}^1\) with moving ramificationhttps://zbmath.org/1521.140542023-11-13T18:48:18.785376Z"Lian, Carl"https://zbmath.org/authors/?q=ai:lian.carl"Moschetti, Riccardo"https://zbmath.org/authors/?q=ai:moschetti.riccardoSummary: Let \(X \to \mathbb{P}^1\) be a general cyclic cover. We give a simple formula for the number of equivariant meromorphic functions on \(X\) subject to ramification conditions at variable points. This generalizes and gives a new proof of a recent result of the second author and \textit{G. P. Pirola} [Isr. J. Math. 249, No. 1, 477--500 (2022; Zbl 1498.14078)] on hyperelliptic odd covers.Moduli spaces of complex affine and dilation surfaceshttps://zbmath.org/1521.140552023-11-13T18:48:18.785376Z"Apisa, Paul"https://zbmath.org/authors/?q=ai:apisa.paul"Bainbridge, Matt"https://zbmath.org/authors/?q=ai:bainbridge.matt"Wang, Jane"https://zbmath.org/authors/?q=ai:wang.janeLes surfaces plates ont souvent désigné les surfaces de translation, comme par exemple dans [\textit{A. Zorich}, in: Frontiers in number theory, physics, and geometry I. On random matrices, zeta functions, and dynamical systems. Papers from the meeting, Les Houches, France, March 9--21, 2003. Berlin: Springer. 437--583 (2006; Zbl 1129.32012)]. Toutefois, ce concept est plus général et contient en particulier les surfaces avec une structure de dilatation ou une structure affine. Cela avait déjà été remarqué par Veech dans [\textit{W. A. Veech}, Am. J. Math. 115, No. 3, 589--689 (1993; Zbl 0803.30037)].
Le présent article précise et étend certains résultats de Veech. Ce faisant il pose de solides bases pour l'étude des espaces des modules des surfaces affines et de dilatations avec des singularités coniques d'angles fixés. En particulier, il détermine les composantes connexes et donne certaines informations sur le groupe fondamental de ces espaces dans le cas des surfaces de dilatations. Cet article est écrit dans un style très clair, il est très agréable à lire. Enfin son approche algébrique complète l'approche dynamique adoptée par de nombreux textes récents (par exemple [\textit{S. Ghazouani}, Sémin. Théor. Spectr. Géom. 35, 69--107 (2019; Zbl 1508.37060)]).
Reviewer: Quentin Gendron (Ciudad de México)Haupt's theorem for strata of abelian differentialshttps://zbmath.org/1521.140562023-11-13T18:48:18.785376Z"Bainbridge, Matt"https://zbmath.org/authors/?q=ai:bainbridge.matt"Johnson, Chris"https://zbmath.org/authors/?q=ai:johnson.chris-r|johnson.charles-c|johnson.chris-a|johnson.chris-g"Judge, Chris"https://zbmath.org/authors/?q=ai:judge.christopher-m"Park, Insung"https://zbmath.org/authors/?q=ai:park.insungLet \(S\) be an oriented unbordered connected surface of genus \(g \geq 2\). A character \(\chi : H_1(S;\mathbb{Z}) \rightarrow \mathbb{C}\) is said to be realized by a complex-valued 1-form \(\omega\) if and only if for each integral cycle \(\gamma\), \(\int_{\gamma} \omega = \chi(\gamma)\). Back in 1920, \textit{O. Haupt} in [Math. Zeitschr. 6, 219--237 (1920; JFM 47.0351.02)] obtained the characters that are realized by some 1-form holomorphic with respect to some complex structure on \(S\). A century later, \textit{M. Kapovich} rewrote the characterization by Haupt in [Contemp. Math. 744, 297--315 (2020; Zbl 1436.37047)].
In the paper under review, the authors give a refinement of Haupt's theorem. Given a holomorphic 1-form \(\omega\), let \(Z(\omega) = \{z_1, \dots, z_k\}\) be the set of zeros of \(\omega\), and \(\alpha_i\) the multiplicity of the zero \(z_i\). Then, the divisor data of \(\omega\), \(\alpha(\omega)\), is the \(n\)-tuple \((\alpha_1, \dots, \alpha_k)\), which satisfies \(\alpha_1 + \dots + \alpha_k = 2g-2\). The main result of the paper determines when a character \(\chi\) is realized by a 1-form \(\omega\) in terms of the divisor data of \(\omega\).
Another proof of this result was independently obtained by \textit{T. Le Fils} in [Int. Math. Res. Not. 2022, No. 8, 5601--5616 (2022; Zbl 1485.30014)].
Reviewer: José Javier Etayo (Madrid)Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials. With an appendix by Gaëtan Borot, Alessandro Giacchetto and Danilo Lewanskihttps://zbmath.org/1521.140572023-11-13T18:48:18.785376Z"Chen, Dawei"https://zbmath.org/authors/?q=ai:chen.dawei"Möller, Martin"https://zbmath.org/authors/?q=ai:moller.martin"Sauvaget, Adrien"https://zbmath.org/authors/?q=ai:sauvaget.adrien\textit{D. Chen} et al. computed the Masur-Veech volume of the moduli space of abelian differentials with fixed distribution of zeros on Riemann surfaces of genus \(g\) [Invent. Math. 222, 283--373 (2020; Zbl 1446.14015)]. Here the authors conjecture a formula for Masur-Veech volumes of the moduli space of quadratic differentials with fixed distribution of zeros on Riemann surfaces of genus \(g\). In more detail, let \((\mu,\nu)\) be an integer partition of \(4g-4\) where \(\mu = (2 m_i)^{r}_{i=1}\) are the even parts and \(\nu = (2 n_j - 1)^s_{j=1}\) are the odd parts and let \(\mathcal Q_{g,r+s} (\mu,\nu)\) be the moduli space parametrizing quadratic differentials \(q\) on Riemann surfaces of genus \(g\) for which \(q\) has \(r\) even-order zeros of type \(\mu\) and \(s\) odd-order zeros of type \(\nu\). Considering the differential up to scaling gives the projectivization \(\mathbb P \mathcal Q_{g,r+s} (\mu,\nu)\) and one can further consider the closure \(\mathbb P \overline{\mathcal Q}_{g,r+s} (\mu,\nu)\) in the incidence variety compactification constructed by \textit{M. Bainbridge} et al. [Algebr. Geom. 6, 196--233 (2019; Zbl 1440.14148)]. Let \(\zeta = c_1 (\mathcal O (1))\) and let \(\psi_i\) denote the cotangent line bundle class on \(\overline{\mathcal M}_{g,n}\) associated with the \(i\)th marked point along with its pullbacks to \(\mathbb P \overline{\mathcal Q}_{g,r+s} (\mu,\nu)\). The authors conjecture the Masur-Veech volume formula
\[
\mathrm{vol}(\mathcal Q_{g,r+s} (\mu,\nu)) = \frac{2^{r-s+3} (2 \pi i)^{2g-2+s}}{(2g-3+r+s)!} \int_{\mathbb P \overline{\mathcal Q}_{g,r+s} (\mu,\nu)} \zeta^{2g+s-3} \psi_1, \dots \psi_r
\]
where \(\psi_i\) are associated with the \(r\) even-order zeros. For \(r=0\) the situation is similar to the case of the minimal strata \(\mathcal H (2g-2)\) of abelian differentials studied by \textit{A. Sauvaget} [Geom. Funct. Anal. 28, 1756--1799 (2018; Zbl 1404.14035)] and thus the authors prove the conjecture in this case. Combining with work of \textit{A. Chiodo} [Compos. Math. 144, 1461--1496 (2008; Zbl 1166.14018)] and \textit{D. Mumford} [Prog. Math. 36, 271--328 (1983; Zbl 0554.14008)], they arrive at a formula for the volume of principal strata of quadratic differentials (those with simples poles and zeros) in terms of the \(\psi_i\) and the top Segre class of the quadratic Hodge bundle on \(\overline{\mathcal M}_{g,k}\). The appendix of Borot, Giacchetto and Lewanski shows how to compute this Segre class by a topological recursion. One consequence is an asympototic growth formula conjectured by \textit{J. E. Andersen} et al. [J. Lond. Math. Soc. 107, 254--332 (2023; Zbl 07730992)], which is especially nice when \(g=1\). The second half of the paper gives applications to Lyaponov exponents and Siegel-Veech constants, proving conjectures due to Grivaux and Hubert and \textit{C. Fougeron} [Math. Res. Lett. 25, No. 4, 1213--1225 (2018; Zbl 1412.37044)]. They conjecture formulas for area Siegal-Veech constants and sums of Lyaponov exponents for general affine invariants submanifolds and prove them for the principal strata. The authors hope to prove the conjecture for all \(r > 0\) in the future.
Reviewer: Scott Nollet (Fort Worth)Boundedness results for 2-adic Galois images associated to hyperelliptic Jacobianshttps://zbmath.org/1521.140582023-11-13T18:48:18.785376Z"Yelton, Jeffrey"https://zbmath.org/authors/?q=ai:yelton.jeffreySummary: Let \(K\) be a number field, and let \(C\) be a hyperelliptic curve over \(K\) with Jacobian \(J\). Suppose that \(C\) is defined by an equation of the form \(y^2 = f (x)(x - \lambda)\) for some irreducible monic polynomial \(f \in \mathcal{O}_K [x]\) of discriminant \(\Delta\) and some element \(\lambda \in \mathcal{O}_K\). Our first main result says that if there is a prime \(\mathfrak{p}\) of \(K\) dividing \(( f(\lambda))\) but not \((2 \Delta)\), then the image of the natural 2-adic Galois representation is open in \(\operatorname{GSp} (T_2 (J))\) and contains a certain congruence subgroup of \(\operatorname{Sp} (T_2 (J))\) depending on the maximal power of \(\mathfrak{p}\) dividing \(( f(\lambda))\). We also present and prove a variant of this result that applies when \(C\) is defined by an equation of the form \(y^2 = f (x) (x - \lambda)(x - \lambda^\prime)\) for distinct elements \(\lambda, \lambda^\prime \in K\). We then show that the hypothesis in the former statement holds for almost all \(\lambda \in \mathcal{O}_K\) and prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.
{{\copyright} 2021 Wiley-VCH GmbH}Computing branches and asymptotes of meromorphic functionshttps://zbmath.org/1521.140592023-11-13T18:48:18.785376Z"Fernández de Sevilla, M."https://zbmath.org/authors/?q=ai:fernandez-de-sevilla.m"Magdalena-Benedicto, R."https://zbmath.org/authors/?q=ai:benedicto.r-magdalena"Pérez-Díaz, S."https://zbmath.org/authors/?q=ai:perez-diaz.soniaThe lowest degree curve \(\tilde{\mathcal{C}}\) approximating a given one \(\mathcal{C}\) near a given point at infinity is known as a generalized asymptote of \(\mathcal{C}\).
The paper begins by recalling the formal definition of a generalized asymptote by means of Puisieux series near the points at infinity of a curve. Later the authors review previously existing algorithms to compute generalized asymptotes of plane algebraic curves, either implicitly or explicitly defined.
As its main result, the paper introduces, by means of the concept of perfect curves (a curve that is its own generalized asymptote), a way to extend those tecniques to parametrically defined curves in any-dimensional spaces through limit computations. It must be also noted that the parametrization is not assumed to be rational.
Reviewer: Jorge Caravantes (Alcalá de Henares)On the disposition of cubic and pair of conics in a real projective plane. IIhttps://zbmath.org/1521.140602023-11-13T18:48:18.785376Z"Gorskaya, Viktoriya Aleksandrovna"https://zbmath.org/authors/?q=ai:gorskaya.viktoriya-aleksandrovnaSummary: The problem of topological classification of real algebraic curves is a classical problem in fundamental mathematics that actually arose at the origins of mathematics. The problem gained particular fame and modern formulation after D. Hilbert included it in his famous list of mathematical problems at number 16 in 1900. This was the problem of classifying curves of the sixth degree, solved in 1969 by \textit{D. A. Gudkov} and \textit{G. A. Utkin} [``Topology of curves of order 6 and surfaces of order 4 (to Hilbert's 16th problem)'' (Russian), Uchen. Zap. Gor'kovsk. Univ., 87, 4--214 (1969), see \textit{D. A. Gudkov}, Transl., Ser. 2, Am. Math. Soc. 112, 9--14 (1978; Zbl 0434.14008) and \textit{G. A. Utkin}, ibid. 123--140 (1978; Zbl 0434.14013)]. In the same place, Gudkov posed the problem of the topological classification of real algebraic curves of degree 6 decomposing into a product of two non-singular curves under certain natural conditions of maximality and general position of quotient curves. Gudkov's problem was solved in [\textit{G. M. Polotovskii}, Sov. Math., Dokl. 18, 1241--1245 (1977; Zbl 0392.14014); translation from Dokl. Akad. Nauk SSSR 236, 548--551 (1977)] and [\textit{I. M. Borisov} and \textit{G. M. Polotovskii}, ``On the topology of planar real decomposable curves of degree 8'' (Russian), Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 176, 3--183 (2020; \url{doi:10.36535/0233-6723-2020-176-3-18})]. At present, after a large series of works by several authors (exact references can be found in [Polotovskii and Borisov, loc. cit.]), the solution of a similar problem on curves of degree 7 is almost complete. In addition, in [\textit{T. V. Kuzmenko} and \textit{G. M. Polotovskiĭ}, ``Classification of curves of degree 6 decomposing into a product of \(M\)-curves in general position'', Transl., Ser. 2, Am. Math. Soc. 173, 165--177 (1996)] a topological classification of curves of degree 6 decomposing into a product of any possible number of irreducible factors in general position, and in [\textit{A. B. Korchagin} and \textit{G. M. Polotovskiĭ}, St. Petersbg. Math. J. 21, No. 2, 231--244 (2010; Zbl 1206.14057); translation from Algebra Anal. 21, No. 2, 92--112 (2009)] a classification of mutual arrangements of \(M\)-quintics, a couple of lines were found.
The present paper is devoted to the case when the irreducible factors of the curve of degree 7 have degrees 3, 2, and 2, and is a continuation of the study begun in [\textit{V. A. Gorskaya} and \textit{G. M. Polotovsky}, ``On the disposition of cubic and pair of conics in a real projective plane'' (Russian), Zh. Sredn. Mat. Obshch. 22, No. 1, 24--37 (2020; \url{doi:10.15507/2079-6900.22.202001.24-37})].On the algebra of elliptic curveshttps://zbmath.org/1521.140612023-11-13T18:48:18.785376Z"Brzeziński, Tomasz"https://zbmath.org/authors/?q=ai:brzezinski.tomaszThe aim of the article is to study all endomorphisms of an elliptic curve \(\mathcal{E}\) that are not necessary isogenies (i.e. they are not required to fix the point at infinity). The author proves that the set of such endomorphisms can be endowed of a \textit{truss} structure. To do that, it first consider the \textit{heap} structure on the points of the curve defined as \([A,B,C]=A-B+C\). This heap structure weakens the group structure of the curve, which can be obtained by choosing a point \(O\) (usually the point at infinity) and taking \([A,O,B]=A+B\). He then shows that the endomorphisms of the curve \(\mathcal{E}\) corresponds to endomorphisms of this heap. In particular, this endomorphisms form a \textit{truss}, a heap equipped with a multiplication law that distributes with respect to the heap law. This truss structure weakens the ring structure on the endomorphisms of \(\mathcal{E}\); moreover, the author notices that a choice of a point \(O\) is needed to recover a ring structure. The construction is given first for a curve defined over \(\mathbb{C}\), then for a generic perfect field.
Reviewer: Lorenzo Furio (Pisa)On group actions on Riemann-Roch spaces of curveshttps://zbmath.org/1521.140622023-11-13T18:48:18.785376Z"Carocca, Angel"https://zbmath.org/authors/?q=ai:carocca.angel"Vásquez Latorre, Daniela"https://zbmath.org/authors/?q=ai:vasquez-latorre.danielaLet \(\mathcal{X}\) be a compact Riemann surface, \(G\) a group of automorphisms of \(\mathcal{X}\) and \(D\) a \(G\)-invariant divisor on \(\mathcal{X}\). Let \(\mathcal{L}(D)\) be the finite dimensional \(\mathbb{C}\)-vector space defined by \[\mathcal{L}(D)=\{f\in\mathbb{C}^*(\mathcal{X})|\mbox{ div}(f)+D\geq0\}\cup\{0\},\] where \(\mbox{div}(f)\) denotes the (principal) divisor of \(f\in\mathbb{C}^*(\mathcal{X})\). The action of \(G\) on \(\mathcal{X}\) induces a linear representation \(L_G(D)\) of \(G\) on the Riemann-Roch space associated to \(D\). The aim of this paper is to determine the structure of \(\mathcal{L}(D)\) as a \(\mathbb{C}[G]\)-module when \(D\) is a \(G\)-invariant divisor; namely, the authors are interested in studying the problem of determining the decomposition of the linear representation \(L_G(D)\) as a sum of complex irreducible representations of \(G\). They give some results on the decomposition of \(L_G(D)\) as sum of complex irreducible representations of \(G\), for \(D\) an effective non-special \(G\)-invariant divisor. In particular, they give explicit formulae for the multiplicity of each complex irreducible factor in the decomposition of \(L_G(D)\) as a sum of complex irreducible representations of \(G\). To illustrate this decomposition the authors give some examples of group actions on Riemann-Roch spaces for divisors on well known families of curves. This paper is organized as follows: Section 1 is an introduction to the subject. Sections 2 and 3 are devoted to group actions on Riemann surfaces and to complex decomposition of \(L_G(D)\), respectively. In Section 4 the authors apply their results to give some examples of group actions on Riemann-Roch spaces for divisors on well known families of curves.
Reviewer: Ahmed Lesfari (El Jadida)On some natural torsors over moduli spaces of parabolic bundleshttps://zbmath.org/1521.140632023-11-13T18:48:18.785376Z"Biswas, Indranil"https://zbmath.org/authors/?q=ai:biswas.indranil|biswas.indranil.1The article under review deals with isomorphism between two natural torsors over moduli spaces of stable parabolic bundles.
Let \(X\) be a compact connected Riemann surface and \(S\) a finite subset. For a rank \(r\) vector bundle we fix parabolic data on \(S\) such that the total parabolic weight for entire \(S\) is an integer. Let \(\mathcal{N}_P(r)\) denote the moduli space of stable parabolic bundles of rank \(r\) and parabolic degree zero with parabolic structure of the given type over the points of \(S\).
Given any stable parabolic bundle \(E_{\ast} \in \mathcal{N}_P(r)\) there is a unique parabolic connection on \(E_{\ast}\) such that the corresponding monodromy homomorphism \(\pi_1 (X \setminus S, x_0 ) \to GL_r(\mathbb{C})\) has its image contained in \(U(r)\) (due to Mehta and Seshadri). The construction of this unitary connection is not algebro-geometric/complex-analytic. Let \(\mathcal{N}C_P(r)\) be the moduli space of pairs of the form \((E_{\ast}, D_E )\), where \(E_{\ast} \in \mathcal{N}_P(r)\) and \(D_E\) is a parabolic connection on \(E_{\ast}\). The projection map defined by \((E_{\ast}, D_E) \mapsto E_{\ast}\) makes \(\mathcal{N}C_P(r)\) a holomorphic torsor over \(\mathcal{N}_P(r)\) for \(T^{\ast}\mathcal{N}_P(r)\). Assigning to every \(E_{\ast} \in \mathcal{N}_P(r)\) the unique parabolic connection \(D_E\) on \(E_{\ast}\) with unitary monodromy, we obtain a section \(\tau_{U P} : \mathcal{N}_P(r) \to \mathcal{N}C_{P}(r)\). This section \( \tau_{U P}\) is not holomorphic.
Let \(\Theta\) be the theta line bundle on \(\mathcal{N}_P(r)\) and let \(\mathcal{U}_P \to \mathcal{N}_P(r)\) be the algebraic fiber bundle defined by the sheaf of holomorphic connections on \(\Theta\). This \(\mathcal{U}_P\) is a holomorphic torsor over \(\mathcal{N}_P(r)\) for \(T^{\ast}\mathcal{N}_P(r)\). We have a Hermitian structure on the line bundle \(\Theta\) due to a construction of Quillen. The corresponding Chern connection on \(\Theta\) defines a \(C^{\infty}\) section \(\tau_{QP} : \mathcal{N}_P(r) \to \mathcal{U}_P\). This section \(\tau_{QP}\) is not holomorphic.
The main result of the article is that there is a natural holomorphic isomorphism between the two \(T^{\ast}\mathcal{N}_P(r)\)-torsors \(\mathcal{N}C_{P}(r)\) and \(\mathcal{U}_P\) that takes the section \(\tau_{U P}\) to the section \(\tau_{QP}\). It is deduced that this isomorphism is in fact algebraic. This result is a parabolic analog of results for vector bundles (see [\textit{I. Biswas} and \textit{J. Hurtubise}, Adv. Math. 389, Article ID 107918, 29 p. (2021; Zbl 1472.14038)]).
Reviewer: Souradeep Majumder (Tirupati)Domination results in \(n\)-Fuchsian fibers in the moduli space of Higgs bundleshttps://zbmath.org/1521.140642023-11-13T18:48:18.785376Z"Dai, Song"https://zbmath.org/authors/?q=ai:dai.song"Li, Qiongling"https://zbmath.org/authors/?q=ai:li.qionglingThe aim of this paper is to show some domination results in \(n\)-Fuchsian fibers in the moduli space of Higgs bundles. The authors show that the energy density of the associated harmonic map of an \(n\)-Fuchsian representation dominates the ones of all other representations in the same Hitchin fiber, which implies the domination of topological invariants: translation length spectrum and entropy. As applications of the energy density domination results, they obtain the existence and uniqueness of equivariant minimal or maximal surfaces in a certain product Riemannian or pseudo-Riemannian manifold. Their proof is based on establishing an algebraic inequality generalizing a theorem of Ness on the nilpotent orbits to general orbits.
This paper is organized as follows: Section 1 is an introduction to the subject and statement of the main result. In Section 2 the authors introduce some facts about Ness' theorem [\textit{L. Ness}, Am. J. Math. 106, 1281--1329 (1984; Zbl 0604.14006)] on the nilpotent orbits generalize this result to the general case. This result plays a key role in this paper and the proof will be postponed to Section 3. Section 4 deals with domination results. In this section, the authors first recall some preliminaries in the non-abelian Hodge theory and higher Teichmüller theory and prove the main theorems. This whole section 5 is devoted to proving a proposition, which plays an important role in the proof of the theorems in the previous section. In Section 6, the authors show some applications of the domination results. They derive two main applications from Sections 4 and 5 to equivariant minimal surfaces and maximal surfaces in product spaces.
Reviewer: Ahmed Lesfari (El Jadida)Meromorphic parahoric Higgs torsors and filtered Stokes G-local systems on curveshttps://zbmath.org/1521.140652023-11-13T18:48:18.785376Z"Huang, Pengfei"https://zbmath.org/authors/?q=ai:huang.pengfei.1|huang.pengfei"Sun, Hao"https://zbmath.org/authors/?q=ai:sun.hao|sun.hao.1|sun.hao.3|sun.hao.2|sun.hao.4The aim of this paper is to consider the problem of generalizing the theory towards obtaining a wild nonabelian Hodge correspondence (NAHC) for a general complex reductive group \(G\). The work deals with the wild NAHC for principal \(G\)-bundles on curves, where \(G\) is a connected complex reductive group. The authors consider the wild NAHC for principal bundles on curves, and prove that there is a one-to-one correspondence between stable meromorphic parahoric Higgs torsors and stable filtered Stokes local systems satisfying a condition introduced by \textit{P. Boalch} [in: Geometry and physics. A festschrift in honour of Nigel Hitchin. Volume 2. Oxford: Oxford University Press. 433--454 (2018; Zbl 1429.58024), Definition 4], which is also referring to the unramified case in the reference [\textit{R. Bezrukavnikov}, ``Non-abelian Hodge moduli spaces and homogeneous affine Springer fibers'', Preprint, \url{arXiv:2209.14695}].
More precisely, they establish the correspondence under a condition on the irregular type of the meromorphic \(G\)-connections introduced by \textit{P. Boalch}, and thus confirm a conjecture [in: Geometry and physics. A festschrift in honour of Nigel Hitchin. Volume 2. Oxford: Oxford University Press. 433--454 (2018; Zbl 1429.58024), subsection 1.5]. They first give a version of Kobayashi-Hitchin correspondence, which induces a one-to-one correspondence between stable meromorphic parahoric Higgs torsors of degree zero (Dolbeault side) and stable meromorphic parahoric connections of degree zero (de Rham side).
Then, by introducing a notion of stability condition on filtered Stokes \(G\)-local systems, they prove a one-to-one correspondence between stable meromorphic parahoric connections of degree zero (de Rham side) and stable filtered Stokes \(G\)-local systems of degree zero (Betti side). When \(G=GL_n(\mathbb{C})\), the main result in this paper reduces to that in [\textit{O. Biquard} and \textit{P. Boalch}, Compos. Math. 140, No. 1, 179--204 (2004; Zbl 1051.53019)]. This paper consists of the following parts:
1. Introduction. This section is an introduction to the subject and statement of the main results.
2. Preliminaries. In this section, the authors briefly review the main algebraic objects and analytic objects studied in this paper.
3. Dolbeault category vs. de Rham category. In this section, the authors introduce the definition of irregular type for meromorphic connections and meromorphic Higgs fields. They give a version of Kobayashi-Hitchin correspondence and obtain a correspondence between \(R\)-stable merohoric Higgs torsors of degree zero and \(R\)-stable merohoric connections of degree zero.
4. de Rham category vs. Betti category. In this section, the authors give the correspondence between merohoric connections and filtered Stokes \(G\)-local systems. By introducing a notion of stability condition for filtered Stokes \(G\)-local systems, they prove a one-to-one correspondence between \(R\)-stable merohoric connections of degree zero and \(R\)-stable filtered Stokes \(G\)-local systems of degree zero.
Reviewer: Ahmed Lesfari (El Jadida)Delta-invariants of complete intersection log del Pezzo surfaceshttps://zbmath.org/1521.140662023-11-13T18:48:18.785376Z"Kim, In-Kyun"https://zbmath.org/authors/?q=ai:kim.in-kyun"Won, Joonyeong"https://zbmath.org/authors/?q=ai:won.joonyeongSummary: We show that complete intersection log del Pezzo surfaces with amplitude one in weighted projective spaces are uniformly \(K\)-stable. As a result, they admit an orbifold Kähler-Einstein metric.On some \(K3\) surfaces with order sixteen automorphismhttps://zbmath.org/1521.140672023-11-13T18:48:18.785376Z"Comparin, Paola"https://zbmath.org/authors/?q=ai:comparin.paola"Priddis, Nathan"https://zbmath.org/authors/?q=ai:priddis.nathan"Sarti, Alessandra"https://zbmath.org/authors/?q=ai:sarti.alessandraSummary: We consider \(K3\) surfaces of Picard rank 14 which admit a purely non-symplectic automorphism of order 16. The automorphism acts on the second cohomology group with integer coefficients and we compute the invariant sublattice for the action. We show that all of these \(K3\) surfaces admit an elliptic fibration and we compute the invariant lattices in a geometric way by using special curves of the elliptic fibration. The computation of these lattices plays an important role when one wants to study moduli spaces and mirror symmetry for lattice polarized \(K3\) surfaces.
{{\copyright} 2022 Wiley-VCH GmbH.}Enriques involutions on pencils of \(K3\) surfaceshttps://zbmath.org/1521.140682023-11-13T18:48:18.785376Z"Festi, Dino"https://zbmath.org/authors/?q=ai:festi.dino"Veniani, Davide Cesare"https://zbmath.org/authors/?q=ai:veniani.davide-cesareSummary: The three pencils of \(K3\) surfaces of minimal discriminant whose general element covers at least one Enriques surface are Kondō's pencils I and II, and the Apéry-Fermi pencil. We enumerate and investigate all Enriques surfaces covered by their general elements.
{{\copyright} 2022 The Authors. \textit{Mathematische Nachrichten} published by Wiley-VCH GmbH.}Equivariant derived equivalence and rational points on \(K3\) surfaceshttps://zbmath.org/1521.140692023-11-13T18:48:18.785376Z"Hassett, Brendan"https://zbmath.org/authors/?q=ai:hassett.brendan"Tschinkel, Yuri"https://zbmath.org/authors/?q=ai:tschinkel.yuriLet \(X\) and \(Y\) be smooth \(K3\) surfaces over a nonclosed field \(K\). Suppose that \(X\) and \(Y\) are derived equivalent over \(K\), that is, there is an equivalence of bounded derived categories of coherent sheaves \(\Phi: D^b(X)\to D^b(Y)\), as triangulated categories, defined over \(K\). A derived equivalence respects many arithmetic properties. The main concern of this article is to consider whether or not \(X(K)\neq \emptyset\) if and ony if \(Y(K)\neq\emptyset\).
The paper presents some results on this question in a very special case -- isotrivial families of \(K3\) surfaces over the pounctured disc.
Let \(G=G_N\) be a finite cyclic group of order \(N\). Fix projective \(K3\) surfaces \(X\) and \(Y\) over \(\mathbf{C}\) with \(G\)-actions and consider the isotrivial families \(\mathcal{X},\,\mathcal{Y}\,\to \Delta_1:=\mathrm{Spec}(\mathbf{C}((t)))\) with generic fibers \(\mathcal{X}_t,\,\mathcal{Y}_t\) over \(K=\mathbf{C}((t))\).
Theorem. Suppose that \(\mathcal{X}_t\) and \(\mathcal{Y}_t\) admit a derived equivalence \(\Phi: D^b(\mathcal{X}_t))\to D^b(\mathcal{Y}_t))\) over \(K\). If \(\mathcal{X}_t(K)\neq\emptyset\), then \(\mathcal{Y}_t(K)\neq\emptyset\).
Proof is based on the analogy between equivariant geometry and descent for nonclosed fields. Isotrivial families over fields of Lautent series are linked to equivariant geometry, and proof is completed through analysis of fixed points. In particular, proof does not hinge on classification.
Reviewer: Noriko Yui (Kingston)Mukai duality via roofs of projective bundleshttps://zbmath.org/1521.140702023-11-13T18:48:18.785376Z"Kapustka, Michał"https://zbmath.org/authors/?q=ai:kapustka.michal"Rampazzo, Marco"https://zbmath.org/authors/?q=ai:rampazzo.marcoSummary: We investigate a construction providing pairs of Calabi-Yau varieties described as zero loci of pushforwards of a hyperplane section on a roof as described in [\textit{A. Kanemitsu}, Mukai pairs and simple $K$-equivalence'', Preprint, \url{arXiv:1812.05392}]. We discuss the implications of such construction at the level of Hodge equivalence, derived equivalence and \({\mathbb{L}} \)-equivalence. For the case of \(K3\) surfaces, we provide alternative interpretations for the Fourier-Mukai duality in the family of \(K3\) surfaces of degree 12 of [\textit{K. Hulek} (ed.) et al., New trends in algebraic geometry. Selected papers presented at the Euro conference, Warwick, UK, July 1996. Cambridge: Cambridge University Press (1999; Zbl 0913.00032); \textit{S. Mukai}, Lond. Math. Soc. Lect. Note Ser. 264, 311--326 (1999; Zbl 0948.14032)]. In all these constructions, the derived equivalence lifts to an equivalence of matrix factorizations categories.\(\mu_p\)- and \(\alpha_p\)-actions on \(K3\) surfaces in characteristic \(p\)https://zbmath.org/1521.140712023-11-13T18:48:18.785376Z"Matsumoto, Yuya"https://zbmath.org/authors/?q=ai:matsumoto.yuyaSummary: We consider \(\mu_p\)- and \(\alpha_p\)-actions on RDP \(K3\) surfaces (\(K3\) surfaces with rational double point (RDP) singularities allowed) in characteristic \(p > 0\). We study possible characteristics, quotient surfaces, and quotient singularities. It turns out that these properties of \(\mu_p\)- and \(\alpha_p\)-actions are analogous to those of \(\mathbb{Z}/l\mathbb{Z}\)-actions (for primes \(l \neq p\)) and \(\mathbb{Z}/p\mathbb{Z}\)-quotients respectively. We also show that conversely an RDP \(K3\) surface with a certain configuration of singularities admits a \(\mu_p\)- or \(\alpha_p\)- or \(\mathbb{Z}/p\mathbb{Z}\)-covering by a ``\(K3\)-like'' surface, which is often an RDP \(K3\) surface but not always, as in the case of the canonical coverings of Enriques surfaces in characteristic 2.\(\mathbb{Z}_2^2\)-actions on Horikawa surfaceshttps://zbmath.org/1521.140722023-11-13T18:48:18.785376Z"Lorenzo, Vicente"https://zbmath.org/authors/?q=ai:lorenzo.vicenteThe Horikawa surfaces in the title are the minimal surfaces of general type with the property that the Noether inequality fails to be an equality at most by \(1\).
In other words, their canonical volume \(K^2\) equals either \(2\chi-6\) or \(2\chi-5\), \(\chi=\chi\left({\mathcal O}\right)\) being the Euler characteristic of their structure sheaf.
These surfaces were already known to the Italian school of Algebraic Geometry at the beginning of the \(XX^{th}\) century. They were named after Horikawa since he described completely their deformation theory and the corresponding moduli spaces.
The main results of this paper may be summarized as follows: every Horikawa surface may be deformed to a surface whose automorphism group contains two commuting involutions.
Reviewer: Roberto Pignatelli (Trento)Simple fibrations in \((1, 2)\)-surfaceshttps://zbmath.org/1521.140732023-11-13T18:48:18.785376Z"Coughlan, Stephen"https://zbmath.org/authors/?q=ai:coughlan.stephen"Pignatelli, Roberto"https://zbmath.org/authors/?q=ai:pignatelli.robertoAuthors' abstract: We introduce the notion of a simple fibration in \((1, 2)\)-surfaces -- that is, a hypersurface inside a certain weighted projective space bundle over a curve such that the general fibre is a minimal surface of general type with \(p_g= 2\) and \(K^2 = 1\). We prove that almost all Gorenstein simple fibrations over the projective line with at worst canonical singularities are canonical threefolds on the Noether line with \(K^3 = \frac{4}{3}p_g- \frac{10}{3}\), and we classify them. Among them, we find all the canonical threefolds on the Noether line that have previously appeared in the literature. The Gorenstein simple fibrations over \(P^1\) are Cartier divisors in a toric \(4\)-fold. This allows to us to show, among other things, that the previously known canonical threefolds on the Noether line form an open subset of the moduli space of canonical threefolds, that the general element of this component is a Mori Dream Space and that there is a second component when the geometric genus is congruent to \(6\) modulo \(8\); the threefolds in this component are new.
Reviewer: Jin-Xing Cai (Beijing)On the complex affine structures of SYZ fibration of del Pezzo surfaceshttps://zbmath.org/1521.140742023-11-13T18:48:18.785376Z"Lau, Siu-Cheong"https://zbmath.org/authors/?q=ai:lau.siu-cheong"Lee, Tsung-Ju"https://zbmath.org/authors/?q=ai:lee.tsung-ju"Lin, Yu-Shen"https://zbmath.org/authors/?q=ai:lin.yu-shenThe authors study Strominger-Yau-Zaslow conjecture [\textit{A. Strominger} et al., Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)] in the case of of special Lagrangian fibration of del Pezzo surface. As main results, they show that the complex affine structure of the special Lagrangian fibration of \(\mathbb{P}^2 \setminus E\), where \(E\) is any given smooth cubic curve in \(\mathbb{P}^2\), constructed by \textit{T. C. Collins} et al. [Duke Math. J. 170, No. 7, 1291--1375 (2021; Zbl 1479.14046)] coincides with the affine structure used in [\textit{M. Carl} et al., ``A tropical view on Landau-Ginzburg models'', unpublished (preliminary version) (2011)]. Moreover,they use the Floer-theoretical gluing method to construct a mirror using immersed Lagrangians, which exactly agrees with the mirror constructed by Carl-Pomperla-Siebert above.
Reviewer: Quanting Zhao (Wuhan)Framed duality and mirror symmetry for toric complete intersectionshttps://zbmath.org/1521.140752023-11-13T18:48:18.785376Z"Rossi, Michele"https://zbmath.org/authors/?q=ai:rossi.micheleSummary: This paper is devoted to systematically extend \(f\)-mirror symmetry between families of hypersurfaces in complete toric varieties, as introduced in [\textit{M. Rossi}, Adv. Theor. Math. Phys. 26, No. 5, 1449--1541 (2022; Zbl 07673666)], to families of complete intersections subvarieties. Namely, \(f\)-mirror symmetry is induced by framed duality of framed toric varieties extending Batyrev-Borisov polar duality between Fano toric varieties. Framed duality has been defined and essentially well described for families of hypersurfaces in toric varieties in the previous [loc. cit.]. Here it is developed for families of complete intersections, allowing us to strengthen some previous results on hypersurfaces. In particular, the class of projective complete intersections and their mirror partners are studied in detail. Moreover, a (generalized) Landau-Ginzburg/Complete-Intersection correspondence is discussed, extending to the complete intersection setup the LG/CY correspondence firstly studied by \textit{A. Chiodo} and \textit{Y. Ruan} [Adv. Math. 227, No. 6, 2157--2188 (2011; Zbl 1245.14038)] and \textit{M. Krawitz} [FJRW rings and Landau-Ginzburg mirror symmetry. Ann Arbor, MI: University of Michigan (Ph.D. Thesis) (2010)].On the liftability of the automorphism group of smooth hypersurfaces of the projective spacehttps://zbmath.org/1521.140762023-11-13T18:48:18.785376Z"González-Aguilera, Víctor"https://zbmath.org/authors/?q=ai:gonzalez-aguilera.victor"Liendo, Alvaro"https://zbmath.org/authors/?q=ai:liendo.alvaro"Montero, Pedro"https://zbmath.org/authors/?q=ai:montero.pedro-jAs the title of this article suggests, the objects studied here are smooth projective hypersurfaces over \(\mathbb{C}\), that is, the zeros of a homogeneous polynomial \(F\) in projective space. Let \(X\) be such a hypersurface, of dimension \(n\geq 1\) and of degree \(d\geq 3\). Under the assumption that \((n,d)\neq (1,3), (2,4)\), it is well-known that the automorphism group \(\mathrm{Aut}(X)\) can be seen as a subgroup of \(\mathrm{PGL}_{n+2}(\mathbb{C})\). The question on ``liftability'' is simply whether this group can be lifted to a subgroup of \(\mathrm{GL}_{n+2}(\mathbb{C})\), isomorphic to the original. A finer version asks whether the form \(F\) stays invariant by such a lift (it could change by a scalar in general). This is what the authors call an \(F\)-lifting, inspired by work of \textit{K. Oguiso} and \textit{X. Yu} [Asian J. Math. 23, No. 2, 201--256 (2019; Zbl 1433.14035)] and \textit{L. Wei} and \textit{X. Yu} [J. Math. Soc. Japan 72, No. 4, 1327--1343 (2020; Zbl 1460.14091)].
The main result of the article (Theorem 3.5) states that, if \(n\) and \(d+2\) are relatively prime, then \(\mathrm{Aut}(X)\) is \(F\)-liftable for \textit{every} \(X\). Moreover, the converse is true in the sense that, if \(n\) and \(d+2\) have a common divisor, then one can always find a hypersurface, defined by a form \(F\), of dimension \(n\) and degree \(d\), whose automorphism group is not \(F\)-liftable. A precise example is given by the Klein hypersurface, defined by the homogeneous form
\[
K=x_0^{d-1}x_1+x_1^{d-1}x_2+\cdots + x_n^{d-1}x_{n+1}+x_{n+1}^{d-1}x_0.
\]
This hypersurface admits automorphisms that are liftable, yet not \(K\)-liftable, and this is the crucial fact used in order to contradict the liftability of the whole automorphism group by some elementary considerations on the determinant.
The positive implication of the main result involves lifting first the \(p\)-Sylow subgroups of \(\mathrm{Aut}(X)\). One then proves that the subgroup generated by these liftings is indeed a lifting of \(\mathrm{Aut}(X)\). Thus, the authors turn their attention first to the question about lifting automorphisms of order \(p^r\) for some prime \(p\). This is addressed in Theorem 2.1, where the authors prove that \(p^r\) is the order of an \(F\)-liftable automorphism for \textit{every} smooth hypersurface of dimension \(n\) and degree \(d\) if and only if one of the following holds:
\begin{itemize}
\item[1.] \(p\) divides \(d-1\) and \(r\leq k(n+1)\), where \(k\) is the \(p\)-adic valuation of \(d-1\);
\item[2.] \(p\) divides \(d\) and \((1-d)^\ell\equiv 1\pmod{p^r}\) for some \(1\leq \ell\leq n+1\);
\item[3.] \(p\) does not divide \(d(d-1)\) and \((1-d)^\ell\equiv 1\pmod{p^r}\) for some \(1\leq \ell\leq n+2\).
\end{itemize}
This result is used to obtain an explicit description of the orders of automorphisms of smooth cubic hypersurfaces in low dimensions (up to \(n=5\)) and smooth quartic threefolds.
Finally, there is a second application of all these results in the last section of the article, giving a criterion for the order of \(\mathrm{Aut}(X)\) not to be divisible by \(p^2\) (Proposition 4.1).
It is worth mentioning that a second article dealing with similar questions [\textit{Z. Zheng}, Isr. J. Math. 247, No. 1, 479--498 (2022; Zbl 1499.14068)] appeared on the arXiv almost at the same time as this one. The authors explain the relations between the two papers in the introduction.
It is also worth mentioning that there is a misprint in the published version of this article (corrected in the final arXiv version, as communicated to me by an author of the paper). It concerns the statement of Lemma 1.6 and its application in Lemma 1.8.
Reviewer: Giancarlo Lucchini Arteche (Santiago)Jordan property for automorphism groups of compact spaces in Fujiki's class \(\mathcal{C} \)https://zbmath.org/1521.140772023-11-13T18:48:18.785376Z"Meng, Sheng"https://zbmath.org/authors/?q=ai:meng.sheng"Perroni, Fabio"https://zbmath.org/authors/?q=ai:perroni.fabio"Zhang, De-Qi"https://zbmath.org/authors/?q=ai:zhang.de-qiSummary: Let \(X\) be a compact complex space in Fujiki's Class \(\mathcal{C} \). We show that the group \(\operatorname{Aut}(X)\) of all biholomorphic automorphisms of \(X\) has the Jordan property: there is a (Jordan) constant \(J = J(X)\) such that any finite subgroup \(G\leqslant \operatorname{Aut}(X)\) has an abelian subgroup \(H\leqslant G\) with the index \([G:H]\leqslant J\). This extends, with a quite different method, the result of \textit{Yu. G. Prokhorov} and \textit{K. A. Shramov} [Math. Notes 106, No. 4, 651--655 (2019; Zbl 1433.14039); translation from Mat. Zametki 106, No. 4, 636--640 (2019)] for Moishezon threefolds.
{{\copyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}Actions of \(\mu_p\) on canonically polarized surfaces in characteristic \(p>0\)https://zbmath.org/1521.140782023-11-13T18:48:18.785376Z"Tziolas, Nikolaos"https://zbmath.org/authors/?q=ai:tziolas.nikolaosAuthor's abstract: This paper studies the existence of non trivial \(\mu_p\) actions on a canonically polarized surface \(X\) defined over an algebraically closed field of characteristic \(p > 0\). In particular, an explicit function \(f (K^2_X )\) is obtained such that if \(p > f (K^2_X )\), then there does not exist a non trivial \(\mu_p\)-action on \(X\). This implies that the connected component of \(\mathrm{Aut}(X)\) containing the identity is either smooth or is obtained by successive extensions by \(\alpha_p\).
Reviewer: Jin-Xing Cai (Beijing)Globally generated vector bundles with small \(c_1\) on projective spaces. IIhttps://zbmath.org/1521.140792023-11-13T18:48:18.785376Z"Anghel, Cristian"https://zbmath.org/authors/?q=ai:anghel.cristian-viorel"Coandă, Iustin"https://zbmath.org/authors/?q=ai:coanda.iustin"Manolache, Nicolae"https://zbmath.org/authors/?q=ai:manolache.nicolaeSummary: We complete the classification of globally generated vector bundles with \(c_1 \leq 5\) on projective spaces by treating the case \(c_1 = 5\) on \(\mathbb{P}^n, n \geq 4\). It turns out that there are very few indecomposable bundles of this kind: besides some obvious examples there are, roughly speaking, only the (first twist of the) rank 5 vector bundle which is the middle term of the monad defining the Horrocks bundle of rank 3 on \(\mathbb{P}^5\), and its restriction to \(\mathbb{P}^4\). We recall, in an appendix, from one of our previous papers, the main results allowing the classification of globally generated vector bundles with \(c_1 = 5\) on \(\mathbb{P}^3\). Since there are many such bundles, a large part of the main body of the paper is occupied with the proof of the fact that, except for the simplest ones, they do not extend to \(\mathbb{P}^4\) as globally generated vector bundles.
Part I, see [\textit{C. Anghel} et al., Globally generated vector bundles with small \(c_1\) on projective spaces. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1407.14042)]
{{\copyright} 2022 Wiley-VCH GmbH.}Projective manifolds whose tangent bundle is Ulrichhttps://zbmath.org/1521.140802023-11-13T18:48:18.785376Z"Benedetti, Vladimiro"https://zbmath.org/authors/?q=ai:benedetti.vladimiro"Montero, Pedro"https://zbmath.org/authors/?q=ai:montero.pedro-j"Prieto-Montañez, Yulieth"https://zbmath.org/authors/?q=ai:prieto-montanez.yulieth"Troncoso, Sergio"https://zbmath.org/authors/?q=ai:troncoso.sergioConsider a vector bundle \(\mathcal{E}\) defined on a polarized variety denoted as \((X, H)\). The vector bundle \(\mathcal{E}\) is termed Ulrich if, for all values of \(j\) ranging from 1 to the dimension of \(X\), and for all values of \(i\), the cohomology groups \(\mathrm{H}^i(X, \mathcal{E}(-jH))\) vanish.
In the context of this article, the authors investigate the problem of characterizing polarized varieties for which either the tangent bundle or the cotangent bundle possesses the Ulrich property. They establish that the sole polarized varieties for which the tangent bundle exhibits the Ulrich property are the twisted cubic and the Veronese surface. Additionally, the authors prove that the cotangent bundle never satisfies the Ulrich condition. To arrive at these conclusions, they employ a meticulous analysis, using the positivity properties of Ulrich vector bundles and breaking down the problem by considering various cases based on the dimension of the polarized variety.
Reviewer: Özhan Genç (Kraków)Tannakian reconstruction of reductive group schemeshttps://zbmath.org/1521.140812023-11-13T18:48:18.785376Z"Zhao, Yifei"https://zbmath.org/authors/?q=ai:zhao.yifeiSummary: We give sharp criteria for when a reductive group scheme satisfies Tannakian reconstruction. When the base scheme is Noetherian, we explicitly identify its Tannaka group scheme.A regular interpolation problem and its applicationshttps://zbmath.org/1521.140822023-11-13T18:48:18.785376Z"Das, Nilkantha"https://zbmath.org/authors/?q=ai:das.nilkanthaLet \(k\) be an algebraically closed field of characteristic zero and \(\Phi : X \longrightarrow Y\) a regular map between affine varieties. The article under review explores the problem of interpolation of a function \(f: Y \longrightarrow k\) over \(\text{Im}(\Phi)\) by a regular function where it is known that \(f \circ \Phi: X \longrightarrow k\) is a regular function, i.e., it asks whether there exists a regular function \(g: Y \longrightarrow k\) such that \(f|_{\text{Im}(\Phi)} = g|_{\text{Im}(\Phi)}\). The author shows that the problem has an affirmative answer if \(Y\) is factorial and \(\Phi\) is almost surjective. The achieved result generalizes a result of \textit{E. Aichinger} [J. Commut. Algebra 7, No. 3, 303--315 (2015; Zbl 1330.13009)] proven for affine spaces.
As an application, the author establishes that a regular morphism from an affine variety to a factorial affine variety is biregular if and only if it is injective and almost surjective, which generalizes a result of \textit{J. Ax} [Pac. J. Math. 31, 1--7 (1969; Zbl 0194.52001)] establishing injective endomorphisms are automorphisms. The author also discusses an analytic criterion of biregularity.
Reviewer: Prosenjit Das (Thiruvananthapuram)Euler-symmetric complete intersections in projective spacehttps://zbmath.org/1521.140832023-11-13T18:48:18.785376Z"Luo, Zhijun"https://zbmath.org/authors/?q=ai:luo.zhijunSummary: Euler-symmetric projective varieties, introduced by Baohua Fu and Jun-Muk Hwang in 2020, are nondegenerate projective varieties admitting many \(\mathbb{C}^{\times}\)-actions of Euler type. They are quasi-homogeneous and uniquely determined by their fundamental forms at a general point. In this paper, we study complete intersections in projective spaces which are Euler-symmetric. It is proven that such varieties are complete intersections of hyperquadrics and the base locus of the second fundamental form at a general point is again a complete intersection.Applications of homogeneous fiber bundles to the Schubert varietieshttps://zbmath.org/1521.140842023-11-13T18:48:18.785376Z"Can, Mahir Bilen"https://zbmath.org/authors/?q=ai:can.mahir-bilen"Saha, Pinaki"https://zbmath.org/authors/?q=ai:saha.pinakiSummary: This article explores the relationship between Schubert varieties and equivariant embeddings, using the framework of homogeneous fiber bundles over flag varieties. We show that the homogenous fiber bundles obtained from Bott-Samelson-Demazure-Hansen varieties are always toroidal. Furthermore, we identify the wonderful varieties among them. We give a short proof of a conjecture of Gao, Hodges, and Yong for deciding when a Schubert variety is spherical with respect to an action of a Levi subgroup. By using BP-decompositions, we obtain a characterization of the smooth spherical Schubert varieties. Among the other applications of our results are: (1) a characterization of the spherical Bott-Samelson-Demazure-Hansen varieties, (2) an alternative proof of the fact that, in type A, every singular Schubert variety of torus complexity 1 is a spherical Schubert variety, and (3) a proof of the fact that, for simply laced algebraic groups of adjoint type, every spherical \(G\)-Schubert variety is locally rigid, that is to say, the first cohomology of its tangent sheaf vanishes.A new approach to the generalized Springer correspondencehttps://zbmath.org/1521.140852023-11-13T18:48:18.785376Z"Graham, William"https://zbmath.org/authors/?q=ai:graham.william-a"Precup, Martha"https://zbmath.org/authors/?q=ai:precup.martha-e"Russell, Amber"https://zbmath.org/authors/?q=ai:russell.amberLet \(G\) be a semisimple algebraic group over \(\mathbb{C}\) and let \(\mathcal{N}\) denote the corresponding nilpotent cone. The Springer resolution \(\tilde{\mathcal{N}} \rightarrow \mathcal{N}\) was used by Borho and MacPherson to prove the Springer correspondence. Indeed, by applying the decomposition theorem to the derived pushforward by the Springer resolution of the constant sheaf, one can recover a correspondence between irreducible representations of the Weyl group and some of the perverse sheaves on \(\mathcal{N}\). The generalized Springer correspondence states that to obtain all the simple perverse sheaves on \(\mathcal{N}\), one must also consider representations of certain relative Weyl groups associated to cuspidal data. In this paper, the authors study a generalization of the method used by Borho and MacPherson and illuminate the generalized Springer correspondence in type A. Namely, they use the decomposition theorem to study the derived pushforward of the constant sheaf under the extended Springer resolution \(\psi\colon \tilde{\mathcal{M}}\rightarrow \tilde{\mathcal{N}}\rightarrow \mathcal{N}\), which was defined by one of the authors in previous work as a composition of the usual Springer resolution. They show it decomposes as a direct sum of all the Lusztig sheaves, which provde the desired correspondence.
In Section 2, the authors provide ample background to the problem at hand. In particular, they explain that in the type A scenario with \(G=\operatorname{SL}(n, \mathbb{C})\), all Lusztig sheaves can be described by characters of \(Z(G)\cong \mathbb{Z}/n\mathbb{Z}\). In Section 3, the authors first prove results about the intersection complexes appearing in the decomposition of the pushforward of the constant sheaf along a finite quotient map. After some analysis, they are able to apply these results to the first map in the extended Springer resolution \(\tilde{\eta}\colon \tilde{\mathcal{M}}\rightarrow \tilde{\mathcal{N}}\), obtaining the formula
\[
\tilde{\eta}_*\underline{\mathbb{C}}_{\tilde{\mathcal{M}}}[\dim \mathcal{N}] = \bigoplus_{\chi \in \widehat{Z}} IC(\tilde{\mathcal{N}}, \mathcal{L}_\chi).
\]
At this point one already sees the new information, with respect to the Borho-MacPherson approach, being considered on \(\tilde{\mathcal{N}}\). In Section 4, the authors factorize the Springer resolution by \(\tilde{\mathcal{N}}\rightarrow \tilde{\mathcal{N}}^P \rightarrow \mathcal{N}\) and prove that the pushforward of these intersection complexes \(IC(\tilde{\mathcal{N}}, \mathcal{L}_\chi)\) along the first map in the composition give \(IC(\tilde{\mathcal{N}^P}, \mathcal{L}_\chi)\) Section 5 completes the analysis and shows that
\[
\tilde{\psi}_*\underline{\mathbb{C}}_{\tilde{\mathcal{M}}}[\dim \mathcal{N}] = \bigoplus_{\chi \in \widehat{Z}} \mathbb{A}_\chi.
\]
Here \(\mathbb{A}_\chi\) ranges over all the Lustzig sheaves, which are known to decompose as a direct sum of representations which provide the bijection assserted by the generalized Springer correspondence.
This new approach the authors take show that it is possible to study the generalized Springer correspondence using the geometry of \(\tilde{\mathcal{M}}\), at least in the type A case. The authors mention that many of these constructions and arguments hold for arbitrary reductive groups, and plan to give a closer analysis of these cases in the future.
The paper is written very clearly and contains many detailed and concrete examples interspersed throughout.
Reviewer: Caleb Ji (New York)Back stable Schubert calculushttps://zbmath.org/1521.140862023-11-13T18:48:18.785376Z"Lam, Thomas"https://zbmath.org/authors/?q=ai:lam.thomas-f"Lee, Seung Jin"https://zbmath.org/authors/?q=ai:lee.seungjin"Shimozono, Mark"https://zbmath.org/authors/?q=ai:shimozono.markIt is a well-known fact in algebraic combinatorics that the (double) Schubert polynomials provide an important family of polynomial representatives for the Schubert classes in the (equivariant) cohomology ring of the flag variety \(\mathrm{Fl}(n)\). In the article under review, the authors develop a theory concerning certain limits of the (double) Schubert polynomials, establishing connections between these limits and the geometry and algebra of symmetric functions. (The authors emphasize that one of their primary motivations for undertaking this work was their investigation of \textit{S. J. Lee}'s recent definition of affine Schubert polynomials [Trans. Am. Math. Soc. 371, No. 6, 4029--4057 (2019; Zbl 1505.14112)].)
There are two closely related infinite-dimensional varieties that play fundamental roles in this paper. Roughly speaking, the \textit{infinite Grassmann variety}, denoted as \(\mathrm{Gr}\), is the union of ordinary Grassmann varieties: \(\bigcup_N \mathrm{Gr}(N,2N)\), where \(\mathrm{Gr}(N,M)\) represents the Grassmann variety of \(N\)-dimensional vector subspaces of the complex vector space \(\mathbb{C}^M\). Let \(\mathrm{Fl}(n)\) denote the flag variety of \(\mathbb{C}^n\). In a similar vein, the \textit{infinite Flag variety}, denoted as \(\mathrm{Fl}\), is the union of ordinary flag varieties: \(\bigcup_N \mathrm{Fl}(2N)\). Much like the role played by Schubert polynomials for the cohomology of the ordinary flag variety, the \textit{back-stable Schubert polynomials} represent Schubert classes in the cohomology of the infinite flag variety. Since their definition is not widely known, we record it here for quick reference:
Let \(S_\mathbb{Z}\) denote the group of permutations of \(\mathbb{Z}\) that move only finitely many integers. For \(w\in S_{\mathbb{Z}}\), let \(\mathfrak{S}_w\) denote the corresponding (Lascoux-Schutzenberger) Schubert polynomial. The \textit{back-stable Schubert polynomial} corresponding to \(w\) is defined as follows:
\[
\overset{\leftarrow}{\mathfrak{S}}_w (x) := \lim_{\substack{p\to-\infty\\ q\to\infty}} \mathfrak{S}_w(x_p, x_{p+1}, \ldots, x_q).
\]
It is worth noting that the definition of the back-stable Schubert polynomials was known to many experts (Buch, Hamaker, Knutson, Lee, Weigandt ..), but the theory surrounding these polynomial gadgets, particularly their relationship to the geometry of infinite flag varieties and double Stanley symmetric functions, had not been fully developed in the literature until this paper emerged.
The \textit{back-stable double Schubert polynomials} can be defined using a limiting procedure as shown in (4.3) in the text. However, here is a convenient sum-of-product description (see Proposition 4.3): \begin{align*}\overset{\leftarrow}{\mathfrak{S}}_w(x;a) = \sum_{w= uv,\ \ell(w)=\ell(u)+\ell(v)} (-1)^{\ell(u)} \overset{\leftarrow}{\mathfrak{S}}_{u^{-1}}(x) \overset{\leftarrow}{\mathfrak{S}}_v(x).\tag{1} \end{align*} Similar to the recursive description of (double) Schubert polynomials in terms of divided difference operators, the back-stable double Schubert polynomials have a recursive description (Theorem 4.7) that makes them a unique family of functions with respect to the initial conditions: \(\overset{\leftarrow}{\mathfrak{S}}_{id}(x;a) = 1\), and \(\overset{\leftarrow}{\mathfrak{S}}_w(a;a) = 0\) if \(w\neq id\).
Now, the \textit{ring of back symmetric formal power series} is defined as follows: \begin{align*}\overset{\leftarrow}{R} := \Lambda \otimes \mathbb{Q}[x_{-1}, x_0, x_1, \ldots], \tag{2}\end{align*} where \(\Lambda\) denotes symmetric functions in variables \(x_{-1}, x_0, \ldots\). Similarly, the \textit{back symmetric double power series ring} is defined as: \begin{align*}\overset{\leftarrow}{R} (x||a) := \Lambda(x||a) \otimes_{\mathbb{Q}} \mathbb{Q}[x_i, a_i \ | \ i\in \mathbb{Z}], \tag{3}\end{align*} where \(\mathbb{Q}[x, a]\) stands for the polynomial ring \(\mathbb{Q}[x_i, a_i \,|\, i \in \mathbb{Z}]\) and \(\Lambda(x||a)\) denotes the ring of double symmetric functions. It is shown in Theorems 3.5 and 4.7 that the back-stable Schubert polynomials (respectively, back-stable Schubert polynomials) form a basis of the ring \(\overset{\leftarrow}{R}\) (respectively, \(\overset{\leftarrow}{R} (x||a)\)).
For \(w\in S_\mathbb{Z}\), let \(F_w\) denote the corresponding Stanley symmetric function. Defining a natural algebra homomorphism \(\eta_0 : \overset{\leftarrow}{R} \to \Lambda\), the authors show that Stanley's definition of \(F_w\) aligns with \(\eta_0(\overset{\leftarrow}{\mathfrak{S}}_w)\). At the core of this observation lies the identity:
\[
\overset{\leftarrow}{\mathfrak{S}}_w = \sum_{w= uv,\ \ell(w)=\ell(u)+\ell(v)} F_u \otimes \mathfrak{S}_v,
\]
which is proven in Theorem 3.14 in the text. All of these concepts extend to the doubled setting, where the authors define the `double Stanley symmetric functions' \(F_w(x||a) \in \Lambda(x||a)\). Subsequently, they expand the eta homomorphism to the doubled setup, denoted as \(\eta_a : \overset{\leftarrow}{R} (x||a)\to \Lambda(x||a)\), thereby establishing that \(\eta_a( \overset{\leftarrow}{R} (x||a) ) = F_w(x||a)\). In fact, toward the end of their article, they introduce the tripled versions of these functions.
For combinatorial interpretations and applications, the authors introduce the concept of a \textit{bumpless pipedream}. In essence, a \textit{bumpless pipedream} is a pipedream where the pipes are not allowed to bump against each other. This new definition enables the authors to explore the monomial expansions of back-stable double Schubert polynomials. By employing bumpless pipedreams, the authors derive numerous intriguing results, including:
\begin{itemize}
\item An expansion for double Schubert polynomials,
\item A positive expression for the coefficient of double Schur functions of Molev in \(\overset{\leftarrow}{\mathfrak{S}}_w(x;a)\), and
\item A novel combinatorial interpretation of Edelman-Greene coefficients.
\end{itemize}
Returning to geometry, in the second half of the article, the authors investigate the equivariant cohomology of the infinite flag variety \(\mathrm{Fl}\) as well as the infinite Grassmannian \(\mathrm{Gr}\). In this context, equivariance pertains to the action of the infinite torus \(T_{\mathbb{Z}}:=\bigcup_{a\leq b} T_{[a,b]}\), where \(T_{[a,b]}\) comprises the elements of \((\mathbb{C}^*)^{\mathbb{Z}}\) with finite support. The cohomology of the classifying space of \(T_{\mathbb{Z}}\) is the polynomial ring \(\mathbb{Q}[a]\), and the fundamental classes of the Schubert varieties in \(\mathrm{Fl}\) constitute a free \(\mathbb{Q}[a]\)-module basis for \(H^{T_{\mathbb{Z}}}_*(\mathrm{Fl})\). In Theorem 6.6, the authors establish that the cohomology \(\mathbb{Q}[a]\)-algebra \(H_{T_{\mathbb{Z}}}^*(\mathrm{Fl})\) (as well as the cohomology \(\mathbb{Q}[a]\)-algebra \(H_{T_{\mathbb{Z}}}^*(\mathrm{Gr})\)) is isomorphic to \(\overset{\leftarrow}{R} (x||a)\) (and \(\Lambda(x||a)\), respectively). These isomorphisms specialize to nonequivariant \(\mathbb{Q}\)-algebra isomorphisms as we omit the role of \(a\)-variables (Theorem 6.7).
In their work [Adv. Math. 62, 187--237 (1986; Zbl 0641.17008)], \textit{B. Kostant} and \textit{S. Kumar} studied the torus-equivariant cohomology of Kac-Moody flag varieties (including the usual flag variety) using the action of the nilHecke ring on these cohomologies. Following a similar path, this current article presents the construction of two commutative actions of the infinite nilHecke ring, denoted as \(\mathbb{A}'\), on the equivariant cohomology \(H_{T_{\mathbb{Z}}}^*(\mathrm{Fl})\), viewed as a \(\mathbb{Q}[a]\)-Hopf algebra. In a related context, \textit{A. I. Molev} [Electron. J. Comb. 16, No. 1, Research Paper R13, 44 p. (2009; Zbl 1182.05128)] defined a basis called the \textit{dual Schur function basis}, denoted \(\hat{s}_\lambda(y||a)\), for the Hopf algebra dual of \(\Lambda(x||a)\). In Proposition 8.1, the authors establish an identification between the Schubert basis of \(H_*^{T^\mathbb{Z}}(\mathrm{Gr})\) and Molev's dual Schur function basis. By introducing homology divided difference operators, they derive a recursive formula for the dual Schur functions \(\hat{s}_\lambda(x||a)\) (Theorem 8.6). Furthermore, the authors compute the ring structure of the equivariant homology ring by proving a positive Pieri rule (Theorem 8.18).
Let \(\widetilde{\mathrm{Gr}}_n\) denote the affine Grassmannian \(\mathrm{SL}(n, \mathbb{C}(\!(t)\!))/\mathrm{SL}(n,\mathbb{C}[\![t]\!])\). Let \(\widetilde{\mathbb{A}}\) denote the affine nilHecke algebra. In his study of the quantum cohomology ring of flag varieties, Peterson showed that the torus equivariant homology of \(\widetilde{\mathrm{Gr}}_n\) is isomorphic to a certain centralizer subalgebra of \(\widetilde{\mathbb{A}}\), denoted \(\widetilde{\mathbb{P}}\). Peterson's work and more are well-explained in \textit{T. Lam}'s paper [J. Am. Math. Soc. 21, No. 1, 259--281 (2008; Zbl 1149.05045)]. In a manner reminiscent of the construction of \(\widetilde{\mathbb{P}}\), the authors here establish the construction of a commutative and cocommutative Hopf subalgebra, denoted as \({\mathbb{P}}'\), within the completed infinite nilHecke algebra \(\mathbb{A}'\). (The authors conjecture that \({\mathbb{P}}'\) is isomorphic to the centralizer of \(\mathbb{Q}[a]\) in \(\mathbb{A}'\).) Furthermore, in Theorem 9.10, they show that \(\mathbb{P}'\) is isomorphic to the \(\mathbb{Q}[a]\)-Hopf algebra \(\hat{\Lambda}(y||a)\), which is the Hopf algebra dual to \(\Lambda(y||a)\), via an explicitly defined map.
Let \(\nu_{a,b} : \Lambda(a) \to \Lambda(b)\) denote the change of variable map from the symmetric functions in \(a\)-variables to the symmetric functions in \(b\)-variables. This map naturally extends to a change of variables map \(\nu_{a,b}: \overset{\leftarrow}{R} (a)\to \Lambda (b) \otimes_{\mathbb{Q}} \mathbb{Q}[a]\). For \(w\in S_{\mathbb{Z}}\), the **back-stable triple Schubert polynomial** is defined as:
\[
\overset{\leftarrow}{\mathfrak{S}}_w (x;a;b) = \nu_{a,b}(\overset{\leftarrow}{\mathfrak{S}}_w(x;a)).
\]
Similarly, the \textit{triple Stanley symmetric function} is defined as:
\[
F_w(x || a || b) := \nu_{a,b}(F_w(x||a)).
\]
The authors show that the formalism they developed for the double Stanley symmetric functions in relation to the back-stable double Schubert polynomials extends to the tripled setting. Utilizing this formalism, they elucidate the computation of some double (and some triple) Edelman-Greene coefficients, as well as the structure constants for dual Schur functions.
It is fair to say that the geometric aspects of triple Stanley symmetric functions are not fully explored in this text. None the less, the authors define another set of Stanley symmetric functions which have a beautiful geometric application. To this end, they consider the following \(\mathbb{Q}[a]\)-algebra homomorphism: \begin{align*} \Lambda(x||a) \otimes_{\mathbb{Q}[a]} \mathbb{Q}[x,a] \to \Lambda(x||a),\tag{4} \end{align*} which is identity on \(\Lambda(x||a)\) and sends \(x_i\) to \(a_{i-n}\). They apply this map to the back stable double Schubert polynomial, \(\overset{\leftarrow}{\mathfrak{S}}_w (x;a)\). The image is called the \textit{\(n\)-rotated double Stanley symmetric function}, denoted \(F^{(n)}_w(x||a)\). The authors show that the \(n\)-rotated double Stanley symmetric function s have a direct geometric interpretation.
It is fair to say that the geometric aspects of triple Stanley symmetric functions are not fully explored in this text. Nonetheless, the authors introduce another set of Stanley symmetric functions with a beautiful geometric application. To achieve this, they consider the following \(\mathbb{Q}[a]\)-algebra homomorphism: \begin{align*} \Lambda(x||a) \otimes_{\mathbb{Q}[a]} \mathbb{Q}[x,a] \to \Lambda(x||a),\tag{5} \end{align*} which is the identity on \(\Lambda(x||a)\) and maps \(x_i\) to \(a_{i-n}\). They apply this map to the back-stable double Schubert polynomial, \(\overset{\leftarrow}{\mathfrak{S}}_w (x;a)\). The resulting image is termed the \textit{\(n\)-rotated double Stanley symmetric function}, denoted as \(F^{(n)}_w(x||a)\). The authors show that the \(n\)-rotated double Stanley symmetric functions possess a direct geometric interpretation that we proceed to review.
The Grassmannian \(\mathrm{Gr}(n,2n)\) can be realized as the quotient \(\pi : M^\circ_{n\times 2n} \to \mathrm{Gr}(n,2n)\), where \(M^\circ_{n\times 2n}\) is the open subset of the affine space consisting of all \(n\times 2n\) complex matrices. For \(w\in S_n\), the \textit{graph Schubert variety}, denoted \(G(w)\), is the Zariski closure \(\overline{\pi (V_w^\circ)}\subset \mathrm{Gr}(n,2n)\), where \(V_w^\circ\) consists of matrices of the form \((Id_n | b_1wb_2)\), where \(b_1\) (resp. \(b_2\)) is an \(n\times n\) complex invertible lower triangular (resp. upper triangular) matrix. In Theorem 12.3, the authors show that under the map (4), the image of \(F^{(n)}_w(x||a)\) in \(H^*_{T_{2n}}(\mathrm{Gr}(n,2n))\) is equal to the class of \(G(w)\). Finally, in Theorem 12.5, they discuss an explicit formula, attributed to Allen Knutson, for the class of \(G(w)\). Since the article is rather long (almost 80 pages), we thought that, for navigation purposes, the inclusion of a table of contents would be useful. Here it is:
\begin{itemize}
\item Introduction
\begin{itemize}
\item[1.1] Flag varieties and Schubert polynomials
\item[1.2] Back stable Schubert polynomials
\item[1.3] Coproduct formula
\item[1.4] Double Stanley symmetric functions
\item[1.5] Bumpless pipedreams
\item[1.6] Infinite flag variety
\item[1.7] Localization and infinite nilHecke algebra
\item[1.8] Homology
\item[1.9] Affine Schubert calculus
\item[1.10] Peterson subalgebra
\item[1.11] Other directions
\end{itemize}
\item Schubert polynomials
\begin{itemize}
\item[2.1] Notation
\item[2.2] Schubert polynomials
\item[2.3] Double Schubert polynomials
\item[2.4] Double Schubert polynomials into single
\item[2.5] Left divided differences
\end{itemize}
\item Back stable Schubert polynomials
\begin{itemize}
\item[3.1] Symmetric functions in nonpositive variables
\item[3.2] Back symmetric formal power series
\item[3.3] Back stable limit
\item[3.4] Stanley symmetric functions
\item[3.5] Negative Schubert polynomials
\item[3.6] Coproduct formula
\item[3.7] Back stable Schubert structure constants
\end{itemize}
\item Back stable double Schubert polynomials
\begin{itemize}
\item[4.1] Double symmetric functions
\item[4.2] Back symmetric double power series
\item[4.3] Localization of back symmetric formal power series
\item[4.4] Back stable double Schubert polynomials
\item[4.5] Double Schur functions
\item[4.6] Double Stanley symmetric functions
\item[4.7] Negative double Schubert polynomials
\item[4.8] Coproduct formula
\item[4.9] Dynkin reversal
\item[4.10] Double Edelman-Greene coefficients
\end{itemize}
\item Bumpless pipedreams
\begin{itemize}
\item[5.1] \(S_{\mathbb{Z}}\)-bumpless pipedreams
\item[5.2] Drooping and the Rothe pipedream
\item[5.3] Halfplane crossless pipedreams
\item[5.4] Rectangular Sn-bumpless pipedreams
\item[5.5] Square \(S_n\)-bumpless pipedreams
\item[5.6] EG pipedreams
\item[5.7] Column moves
\item[5.8] Insertion
\end{itemize}
\item Infinite flag variety
\begin{itemize}
\item[6.1] Infinite Grassmannian
\item[6.2] Infinite flag variety
\item[6.3] Schubert varieties
\item[6.4] Equivariant cohomology of infinite flag variety
\item[6.5] Localization and GKM rings for infinite flags and infinite Grassmannian
\item[6.6] Realization of Schubert basis of GKM ring by backstable Schubert polynomials
\item[6.7] Shifting
\end{itemize}
\item NilHecke algebra and Hopf structure
\begin{itemize}
\item[7.1] NilHecke algebra
\item[7.2] NilHecke actions
\item[7.3] Hopf structure on \(\Psi_{\mathrm{Gr}}\)
\end{itemize}
\item Homology Hopf algebra
\begin{itemize}
\item[8.1] Molev's dual Schur functions
\item[8.2] Homology divided difference operators
\item[8.3] \(\delta\)-Schubert polynomials and \(\delta\)-Schur functions
\item[8.4] \(\delta\)-dual Schurs represent Knutson-Lederer classes
\item[8.5] Homology equivariant Monk's rule
\item[8.6] Homology equivariant Pieri rule
\end{itemize}
\item Peterson subalgebra
\begin{itemize}
\item[9.1] Affine symmetric group
\item[9.2] Translation elements
\item[9.3] The Peterson subalgebra
\item[9.4] Fomin-Stanley algebra
\item[9.5] Stability of affine double Edelman-Greene coefficients
\item[9.6] Proof of Proposition 9.6
\item[9.7] Proof of Theorem 9.7
\item[9.8] Proof of Theorems 9.8 and 9.10
\item[9.9] Proof of Theorem 4.22
\end{itemize}
\item Back stable triple Schubert polynomials
\begin{itemize}
\item[10.1] Tripling
\item[10.2] Back stable triple Schubert polynomials
\item[10.3] Triple Stanley symmetric functions
\item[10.4] Double to triple
\item[10.5] Triple Edelman-Greene coefficients for a hook
\item[10.6] Proof of Theorem 8.15
\item[10.7] Proof of Theorem 8.18
\end{itemize}
\item Affine flag variety
\begin{itemize}
\item[11.1] Affine flag variety and affine Grassmannian
\item[11.2] Equivariant cohomology of affine flag variety
\item[11.3] Presentations
\item[11.4] Small affine Schubert classes
\item
\end{itemize}
\item Graph Schubert varieties
\begin{itemize}
\item[12.1] Schubert varieties and double Schur functions
\item[12.2] The graph Schubert class
\item[12.3] Proof of Theorem 12.3
\item[12.4] Proof of Theorem 5.11
\item[12.5] Divided difference formula for graph Schubert class
\item
\end{itemize}
\item[] Appendix A. Dictionary between positive and nonpositive alphabets
\begin{itemize}
\item[A.1] Positive alphabets
\item[A.2] Nonpositive alphabets
\item[A.3] Localization
\item[A.4] Molev's skew double Schur functions
\end{itemize}
\item[] Appendix B. Schubert Inversion
\begin{itemize}
\item[B.1] Proof of Lemma 2.10
\item[B.2] Inverting systems with Schubert polynomials as change-of-basis matrix
\end{itemize}
\item[] Appendix C. Level zero affine nilHecke ring
\begin{itemize}
\item[C.1] Level zero affine nilHecke ring
\item[C.2] Peterson algebra
\end{itemize}
\end{itemize}
Reviewer: Can Mahir Bilen (New Orleans)An introduction to rationally connected fibrations over curves and surfaceshttps://zbmath.org/1521.140872023-11-13T18:48:18.785376Z"Fanelli, Andrea"https://zbmath.org/authors/?q=ai:fanelli.andreaThis note provides an introduction to some of the main problems in the theory of rationally connected varieties. In the first part, the author deals with the existence of rational sections of fibrations \(X\rightarrow B\).
In particular, when \(B\) is a smooth curve and the general fiber is rationally connected, the fibration exists [\textit{T. Graber} et al., J. Am. Math. Soc. 16, No. 1, 57--67 (2003; Zbl 1092.14063)]. A proof is provided.
After that, the author uses that to prove a result of \textit{A. J. de Jong} et al. [Publ. Math., Inst. Hautes Étud. Sci. 114, 1--85 (2011; Zbl 1285.14053)]. That is the following: Let \(K\) be the function field of a complex surface \(B\), then for any complex algebraic group \(G\) which is connected, simply connected and semisimple over \(K\), any \(G\)-torsor over \(K\) is trivial. This implies that if \(X\rightarrow B\) is a \(G\)-bundle, then it has a rational section.
In the last section, he deals with two possible generalizations of Tsen-Lang numerical condition [\textit{S. Lang}, Ann. Math. (2) 55, 373--390 (1952; Zbl 0046.26202)]. Those are rationally connected varieties and \(2\)-Fano varieties.
Reviewer: Giosuè Muratore (Roma)Gorenstein Fano generic torus orbit closures in \(G/P\)https://zbmath.org/1521.140882023-11-13T18:48:18.785376Z"Montagard, Pierre-Louis"https://zbmath.org/authors/?q=ai:montagard.pierre-louis"Rittatore, Alvaro"https://zbmath.org/authors/?q=ai:rittatore.alvaroSummary: Given a reductive group \(G\) and a parabolic subgroup \(P\subset G\), with maximal torus \(T\), we consider (following Dabrowski's work) the closure \(X\) of a generic \(T\)-orbit in \(G/P\) and determine in combinatorial terms when the toric variety \(X\) is \(\mathbb{Q}\)-Gorenstein Fano, extending in this way the classification of smooth Fano generic closures given by Voskresenskiĭ and Klyachko. As an application, we apply the well-known correspondence between Gorenstein Fano toric varieties and reflexive polytopes in order to exhibit which reflexive polytopes correspond to generic closures -- this list includes the reflexive root polytopes.On the integrals over two-dimensional compact complex toric varietieshttps://zbmath.org/1521.140892023-11-13T18:48:18.785376Z"Ul'vert, Ol'ga S."https://zbmath.org/authors/?q=ai:ulvert.olga-sSummary: In this paper, we proof that an integral of smooth \((2,2)\)-form over two-dimensional compact complex toric variety \(X\) (which contains complex torus \(\mathbb{T}^2)\) is equal to the integral of holomorphic \((2,0)\)-form over real torus \(T^2\subset\mathbb{T}^2\).Classification of Levi-spherical Schubert varietieshttps://zbmath.org/1521.140902023-11-13T18:48:18.785376Z"Gao, Yibo"https://zbmath.org/authors/?q=ai:gao.yibo"Hodges, Reuven"https://zbmath.org/authors/?q=ai:hodges.reuven"Yong, Alexander"https://zbmath.org/authors/?q=ai:yong.alexanderSummary: A Schubert variety in the complete flag manifold \(GL_n/B\) is \textit{Levi-spherical} if the action of a Borel subgroup in a Levi subgroup of a standard parabolic has an open dense orbit. We give a combinatorial classification of these Schubert varieties. This establishes a conjecture of the latter two authors, and a new formulation in terms of standard Coxeter elements. Our proof uses and contributes to the theory of \textit{key polynomials} (type \(A\) Demazure module characters).\(K\)-theory of regular compactification bundleshttps://zbmath.org/1521.140912023-11-13T18:48:18.785376Z"Uma, V."https://zbmath.org/authors/?q=ai:uma.vikramanSummary: Let \(G\) be a split connected reductive algebraic group. Let \(\mathcal{E}\longrightarrow \mathcal{B}\) be a \(G\times G\)-torsor over a smooth base scheme \(\mathcal{B}\) and \(X\) be a regular compactification of \(G\). We describe the Grothendieck ring of the associated fibre bundle \(\mathcal{E}(X):=\mathcal{E}\times_{G\times G} X\), as an algebra over the Grothendieck ring of a canonical toric bundle over a flag bundle over \(\mathcal{B}\). These are relative versions of the corresponding results on the Grothendieck ring of \(X\) in the case when \(\mathcal{B}\) is a point, and generalize the classical results on the Grothendieck rings of projective bundles, toric bundles and flag bundles.
{{\copyright} 2022 Wiley-VCH GmbH.}Geometry of \(\mathrm{SU}(3)\)-character varieties of torus knotshttps://zbmath.org/1521.140922023-11-13T18:48:18.785376Z"González-Prieto, Ángel"https://zbmath.org/authors/?q=ai:gonzalez-prieto.angel"Martínez, Javier"https://zbmath.org/authors/?q=ai:martinez-martinez.javier"Muñoz, Vicente"https://zbmath.org/authors/?q=ai:munoz.vicenteSummary: We describe the geometry of the character variety of representations of the knot group \(\Gamma_{m , n} = \langle x, y | x^n = y^m \rangle\) into the group \(\mathrm{SU}(3)\), by stratifying the character variety into strata corresponding to totally reducible representations, representations decomposing into a 2-dimensional and a 1-dimensional representation, and irreducible representations, the latter of two types depending on whether the matrices have distinct eigenvalues, or one of the matrices has one eigenvalue of multiplicity 2. We describe how the closure of each stratum meets lower strata, and use this to compute the compactly supported Euler characteristic, and to prove that the inclusion of the character variety for \(\mathrm{SU}(3)\) into the character variety for \(\mathrm{SL}(3, \mathbb{C})\) is a homotopy equivalence.Erratum to: ``Transverse lines to surfaces over finite fields''https://zbmath.org/1521.140932023-11-13T18:48:18.785376Z"Asgarli, Shamil"https://zbmath.org/authors/?q=ai:asgarli.shamil"Duan, Lian"https://zbmath.org/authors/?q=ai:duan.lian"Lai, Kuan-Wen"https://zbmath.org/authors/?q=ai:lai.kuan-wenThe proof of Theorem 4.5 in the authors' paper [ibid. 165, No. 1--2, 135--157 (2021; Zbl 1464.14051)] contained a subtle error in the application of Lemma 4.6. Here, a corrected proof is given (which doesn't need Lemma 4.6).Typical labels of real formshttps://zbmath.org/1521.140942023-11-13T18:48:18.785376Z"Ballico, Edoardo"https://zbmath.org/authors/?q=ai:ballico.edoardoSummary: Let \(X(\mathbb{C})\subset \mathbb{P}^r(\mathbb{C})\) be an integral projective variety defined over \(\mathbb{R}\). Let \(\sigma\) denote the complex conjugation. A point \(q\in\mathbb{P}^r(\mathbb{R})\) is said to have \((a,b)\in\mathbb{N}^2\) as a label if there is \(S\subset X(\mathbb{C})\) such that \(\sigma(S)=S\), \(S\) spans \(q\), \(\#S=2a+b\) and \(\#(S\cap X(\mathbb{R}))=b\). We say that \((a,b)\) has weight \(2a+b\). A label-weight \(t\) is typical for the \(k\)-secant variety \(\sigma_k(X(\mathbb{C}))\) of \(X(\mathbb{C})\) if there is a non-empty Euclidean open subset \(V\) of \(\sigma_k(X(\mathbb{C}))(\mathbb{R})\) such that all \(q\in V\) have a label of weight \(t\) and no label of weight \(<t\). The integer \(k\) is always the minimal label-weight of \(\sigma_k(X(\mathbb{C}))(\mathbb{R})\) if \(\sigma_{k-1}(X(\mathbb{C}))\ne\mathbb{P}^r(\mathbb{C})\). In this paper \(X(\mathbb{C})=X_{n,d}(\mathbb{C})\) is the order \(d\) Veronese embedding of \(\mathbb{P}^n(\mathbb{C})\). We prove that \(k\) and \(k+1\) are the typical label-weights of \(\sigma_k(X(\mathbb{C}))(\mathbb{R})\) if \((n,d,k)\in\{(2,6,9),(3,4,8),(5,3,9),(2,4,5),(4,3,7)\}\). These examples are important, because the first 3 are the ones in which generic uniqueness for proper secant varieties fails for the \(k\)-secant variety (a theorem by Chiantini, Ottaviani and Vannieuwenhoven), the fourth is in the Mukai list (fano 3-fold \(V_{22}\)) and the last one appears in the Alexander-Hirschowitz list of exceptional secant varieties of Veronese embeddings.On the cuspidal locus in the dual varieties of Segre quartic surfaceshttps://zbmath.org/1521.140952023-11-13T18:48:18.785376Z"Honda, Nobuhiro"https://zbmath.org/authors/?q=ai:honda.nobuhiro"Minagawa, Ayato"https://zbmath.org/authors/?q=ai:minagawa.ayatoSummary: Motivated by a kind of Penrose correspondence, we investigate the space of hyperplane sections of Segre quartic surfaces which have an ordinary cusp. We show that the space of such hyperplane sections is empty for two kinds of Segre surfaces, and it is a connected surface for all other kinds of Segre surfaces. We also show that when it is non-empty, the closure of the space is either birational to the surface itself or birational to a double covering of the surface, whose branch divisor consists of some specific lines on the surface.On the base locus of linear systems of general double pointshttps://zbmath.org/1521.140962023-11-13T18:48:18.785376Z"Ballico, Edoardo"https://zbmath.org/authors/?q=ai:ballico.edoardoSummary: Fix integers \(n\ge 1\), \(d\ge 4\) and \(x>0\) such that \((n+1)(x-1)+\binom{n+2}{2}\le\binom{n+d}{n}\). Take a general \(S\subset\mathbb{P}^n\) such that \(\#S=x\) and let \(\mathcal{B}\) denote the scheme-theoretic base locus of \(|\mathcal{I}_{2S}(d)|\), where \(2S\) is the union of the double points with \(S\) as their reduction. Then \(2S\) is the union of the connected components of \(\mathcal{B}\) containing at least one point of \(S\). We prove this theorem proving that a general union of \(x-1\) double points and one triple point has no higher cohomology in degree \(d\).D-critical loci for length \(n\) sheaves on local toric Calabi-Yau 3-foldshttps://zbmath.org/1521.140972023-11-13T18:48:18.785376Z"Katz, Sheldon"https://zbmath.org/authors/?q=ai:katz.sheldon"Shi, Yun"https://zbmath.org/authors/?q=ai:shi.yun.2Summary: The notion of a d-critical locus is an ingredient in the definition of motivic Donaldson-Thomas invariants by \textit{V. Bussi} et al. [J. Algebr. Geom. 28, No. 3, 405--438 (2019; Zbl 1441.14059)]. There is a canonical d-critical locus structure on the Hilbert scheme of dimension zero subschemes on local toric Calabi-Yau 3-folds. This is obtained by truncating the \(-1\)-shifted symplectic structure on the derived moduli stack . In this paper we show the canonical d-critical locus structure has critical charts consistent with the description of Hilbert scheme as a degeneracy locus . In particular, the canonical d-critical locus structure is isomorphic to the one constructed in a previous paper by the authors for local \(\mathbb{P}^2\) and local \(\mathbb{F}_n\).
{{\copyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}Algebraic reduced genus one Gromov-Witten invariants for complete intersections in projective spaces. IIhttps://zbmath.org/1521.140982023-11-13T18:48:18.785376Z"Lee, Sanghyeon"https://zbmath.org/authors/?q=ai:lee.sanghyeon"Oh, Jeongseok"https://zbmath.org/authors/?q=ai:oh.jeongseokSummary: In [``Algebraic reduced genus one Gromov-Witten invariants for complete intersections in projective spaces'', Preprint, \url{arXiv:1809.10995}], we provided an algebraic proof of \textit{A. Zinger}'s comparison formula [Geom. Topol. 12, No. 2, 1203--1241 (2008; Zbl 1167.14009); J. Differ. Geom. 83, No. 2, 407--460 (2009; Zbl 1186.53100)] between genus one Gromov-Witten invariants and reduced invariants when the target space is a complete intersection of dimension 2 or 3 in a projective space. In this paper, we extend the proof in [Lee and Oh, loc. cit.] to all dimensions and to descendant invariants.On polynomial images of a closed ballhttps://zbmath.org/1521.140992023-11-13T18:48:18.785376Z"Fernando, José F."https://zbmath.org/authors/?q=ai:fernando.jose-f"Ueno, Carlos"https://zbmath.org/authors/?q=ai:ueno.carlosThe authors characterize semialgebraic subsets of \(\mathbb R^n\) that are the image of a real morphism (i.e. real polynomial map) of a closed unit ball \(\mathbb R^m\). In particular, they proved:
If \(S\subset \mathbb R^n\) is a finite union of \(n\)-dimensional compact and convex polyhedra then the following are equivalent
\begin{itemize}
\item \(S\) is connected by real analytic paths
\item There is a real morphism \(f:\mathbb R^{n+1} \to \mathbb R^n\) such that \(f(B_{n+1})= S\)
\item There is a real morphism \(f:\mathbb R^n \to \mathbb R^n\) such that \(f(B_n)= S\)
\end{itemize}
where \(B_k\subset \mathbb R^k\) is a closed unit ball in the Euclidean topology.
This is a special case of their main result, where \(S\) above can be replaced with a finite union of \textit{\(m\)-bricks} and the morphism is \(f:\mathbb R^{m+1}\to \mathbb R^n\). An \textit{\(m\)-brick} is a set \(T\subset \mathbb R^n\) such that there is a homotopy \[H_\lambda :B_m \to T \quad \lambda \in [0,1]\] that deforms \(H_0(B_m)=T\) to a point \(H_1(B_m)\) and has intermediate images contained in the Euclidean interior of \(T\).
The authors also pose an open problem asking the minimum degrees of such polynomial maps.
Reviewer: Jose Capco (Linz)Patchworking real algebraic hypersurfaces with asymptotically large Betti numbershttps://zbmath.org/1521.141002023-11-13T18:48:18.785376Z"Arnal, Charles"https://zbmath.org/authors/?q=ai:arnal.charlesSummary: In this article, we describe a recursive method for constructing a family of real projective algebraic hypersurfaces in ambient dimension \(n\) from families of such hypersurfaces in ambient dimensions \(k=1,\ldots ,n-1\). The asymptotic Betti numbers of real parts of the resulting family can then be described in terms of the asymptotic Betti numbers of the real parts of the families used as ingredients. The algorithm is based on \textit{O. Viro}'s Patchwork [``Patchworking real algebraic varieties'', Preprint, \url{arXiv:math/0611382}] and inspired by \textit{I. Itenberg} and \textit{O. Viro}'s construction of asymptotically maximal families in arbitrary dimension [in: Proceedings of the 13th Gökova geometry-topology conference, Gökova, Turkey, May 28--June 2, 2006. Cambridge, MA: International Press. 91--105 (2007; Zbl 1180.14055)]. Using it, we prove that for any \(n\) and \(i=0,\ldots ,n-1\), there is a family of asymptotically maximal real projective algebraic hypersurfaces \(\lbrace Y^n_d\rbrace_d\) in \({\mathbb{R}}{\mathbb{P}}^n\) (where \(d\) denotes the degree of \(Y^n_d)\) such that the \(i\) th Betti numbers \(b_i({\mathbb{R}}Y^n_d)\) are asymptotically strictly greater than the \((i,n-1-i)\) th Hodge numbers \(h^{i,n-1-i}({\mathbb{C}}Y^n_d)\). We also build families of real projective algebraic hypersurfaces whose real parts have asymptotic (in the degree \(d)\) Betti numbers that are asymptotically (in the ambient dimension \(n)\) very large.
{{\copyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}The real Plücker-Klein maphttps://zbmath.org/1521.141012023-11-13T18:48:18.785376Z"Krasnov, Vyacheslav A."https://zbmath.org/authors/?q=ai:krasnov.vyacheslav-aSummary: We consider the generalized Plücker-Klein map from the set of all real marked biquadrics to the set of real Kummer varieties. We find a necessary and sufficient condition on a real marked biquadric in order that the corresponding real Kimmer variety be isomorphic to the real Kummer variety induced by the real Jacobian of a double covering of the pencil of quadrics through the given biquadric. We also give a deformation classification of the real Plücker-Klein map.Reflection groups and cones of sums of squareshttps://zbmath.org/1521.141022023-11-13T18:48:18.785376Z"Debus, Sebastian"https://zbmath.org/authors/?q=ai:debus.sebastian"Riener, Cordian"https://zbmath.org/authors/?q=ai:riener.cordianStudying the difference between the Cone of Sums of Squares and the Cone of Nonnegative Real Polynomials is a crucial problem in real algebraic geometry. In this article, the authors focus extensively on the structure of the Sums of Squares Cone, which remains invariant under reflection groups. In particular, they effectively analyze the structure of SoS cones invariant under \(A_n\), \(B_n\), and \(D_n\), utilizing Specht's polynomial representations of reflection group as introduced by \textit{H. Morita} and \textit{H. F. Yamada} [Hokkaido Math. J. 27, No. 3, 505--515 (1998; Zbl 0919.20027)] in their work on Higher Specht Polynomials for the complex reflection group.
Research on when non-negative polynomials can be expressed as sums of squares has a long history, dating back to Hilbert's work on forms with given degrees and the number of variables. Notably, studies have been conducted on symmetric forms, such as Choi, Lam, and Resnick's results on symmetric sextics and Harris's investigations into symmetric ternary octics, where it has been established that non-negative polynomials can indeed be represented as sums of squares.
Extensive research in the invariant theory of finite reflection groups has allowed the generalization of the theories and results from symmetric polynomials. In particular, this paper, focusing on reflection groups that are invariant under \(A_n\), \(B_n\), and \(D_n\), achieves a concrete characterization of the cone of invariant sums of squares in Theorem 3.11.
Using this characterization, in Theorem 4.1, the authors provide an explicit description of the dual cone of even symmetric ternary octic sums of squares, essentially reaffirming Harris's results concerning symmetric ternary octics. Furthermore, in Theorem 4.18, they prove that any nonnegative ternary octics invariant under \(D_3\) can be expressed as a sum of squares. In Theorem 4.22, the authors shed light on the structure of the dual cone of sums of squares of quaternary quartics invariant under \(D_4\), and in Theorem 4.26, they establish whether all nonnegative polynomials for a given \((n,2d)\) are sums of squares invariant under \(D_n\). Finally, the article concludes by highlighting some connections to non-negativity testing of forms using semidefinite programming in the last subsection.
Reviewer: Jaewoo Jung (Daejeon)Computing \(\mathbb{A}^1\)-Euler numbers with Macaulay2https://zbmath.org/1521.141032023-11-13T18:48:18.785376Z"Pauli, Sabrina"https://zbmath.org/authors/?q=ai:pauli.sabrina\(\mathbb{A}^1\)-enumerative geometry is a recent approach to enumerative geometry over non-closed fields. Rather than integer-valued counts, one obtains counts valued in the Grothendieck-Witt ring \(\mathrm{GW}(k)\) of isomorphism classes of symmetric, non-degenerate bilinear forms over the base field. The ring \(\mathrm{GW}(k)\) shows up as the \(\mathbb{A}^1\)-homotopy classes of self-maps of the motivic sphere \(\mathbb{P}^n/\mathbb{P}^{n-1}\) [\textit{F. Morel}, \(\mathbb{A}^1\)-algebraic topology over a field. Berlin: Springer (2012; Zbl 1263.14003)]. Tools like Euler classes and local Brouwer degrees admit analogs in \(\mathbb{A}^1\)-homotopy theory that, despite their homotopical origin, can be computed explicitly in terms of commutative algebra and algebraic geometry [\textit{J. L. Kass} and \textit{K. Wickelgren}, Duke Math. J. 168, No. 3, 429--469 (2019; Zbl 1412.14014); \textit{T. Bachmann} and \textit{K. Wickelgren}, J. Inst. Math. Jussieu 22, No. 2, 681--746 (2023; Zbl 1515.14037)].
In this article, the author provides a clearly written overview of the role of these tools in \(\mathbb{A}^1\)-enumerative geometry. She then demonstrates how one can use the Macaulay2 computer algebra software to compute \(\mathrm{GW}(k)\)-valued Euler numbers and local degrees. This makes computations in \(\mathbb{A}^1\)-enumerative geometry quite accessible. For example, it would be feasible to design a vertically-integrated research project on some \(\mathbb{A}^1\)-enumerative problem, where an undergraduate or early graduate student is tasked with generating some data or exploring examples in Macaulay2. Such a project would give the student concrete computational tasks to work on while they learn the relevant theory in algebraic geometry or homotopy theory.
It is also worth mentioning the forthcoming package \texttt{A1BrouwerDegrees} for Macaulay2, which is being developed by Borisov, Brazelton, Espino, Hagedorn, Han, Lopez Garcia, Louwsma, and Tawfeek. This package will implement the commutative algebraic formula given in [\textit{T. Brazelton} et al., Algebra Number Theory 17, No. 11, 1985--2012 (2023; Zbl pending)] for computing \(\mathbb{A}^1\)-degrees, making the examples in the paper under review even more readily available.
Reviewer: Stephen McKean (Cambridge, MA)Complements of discriminants of real parabolic function singularitieshttps://zbmath.org/1521.141042023-11-13T18:48:18.785376Z"Vassiliev, V. A."https://zbmath.org/authors/?q=ai:vassiliev.victor-a|vassilev.victor-aLet \(f:(\mathbb{R}^n,0)\rightarrow (\mathbb{R},0)\), \(df (0)=0\), be an analytic function singularity, and \(F:(\mathbb{R}^n\times \mathbb{R}^{\ell},0)\rightarrow (\mathbb{R},0)\) be an arbitrary its analytic deformation, i.~e., a family of functions \(f_{\lambda}\equiv F(\cdot ,\lambda )\), \(f_0\equiv f\). The discriminant (or the level bifurcation set) \(\Sigma =\Sigma (f)\) of this deformation is the set of points \(\lambda\) from a neighborhood of the origin in the parameter space \(\mathbb{R}^{\ell}\) such that the corresponding functions \(f_{\lambda}\) have real critical points with zero critical value close to the origin in \(\mathbb{R}^n\). The discriminant is a subvariety in \(\mathbb{R}^{\ell}\) (of codimension 1 in all interesting cases) which can divide a neighborhood of the origin into several connected components. The author gives a conjecturally complete list of components of complements of discriminant varieties of parabolic singularities of smooth real functions. He also promotes a combinatorial program that enumerates topological types of non-discriminant morsifications of isolated real function singularities and provides a strong invariant of the components of complements of discriminant varieties.
Reviewer: Vladimir P. Kostov (Nice)Characterization of tropical plane curves up to genus sixhttps://zbmath.org/1521.141052023-11-13T18:48:18.785376Z"Tewari, Ayush Kumar"https://zbmath.org/authors/?q=ai:tewari.ayush-kumarSummary: We provide a new forbidden criterion for the realizability of smooth tropical plane curves. This in turn provides us a complete classification of smooth tropical plane curves up to genus six.The capacity of quiver representations and Brascamp-Lieb constantshttps://zbmath.org/1521.160092023-11-13T18:48:18.785376Z"Chindris, Calin"https://zbmath.org/authors/?q=ai:chindris.calin"Derksen, Harm"https://zbmath.org/authors/?q=ai:derksen.harmGiven a real representation \(V\) of a bipartite quiver \(Q\), along with an integral weight \(\sigma\) of \(Q\) orthogonal to the dimension vector of \(V\), the authors introduce the Brascamp-Lieb (BL) operator \(T_{V,\sigma}\) associated to this datum, and study its capacity \(D_Q(V,\sigma)\). They characterise the positivity of \(D_Q(V,\sigma)\) by the \(\sigma\)-semi-stability of \(V\) (Theorem 1) and give a character formula in case the capacity is positive (Theorem 2). In the case of the \(m\)-subspace quiver, \(D_Q(V,\sigma)\) is related to the BL constants in the \(m\)-multilinear BL inequality in analysis.
The article is organised in five sections.
In the introduction, the authors motivate their studies by a result and comment in [\textit{J. Bennett} et al., Geom. Funct. Anal. 17, No. 5, 1343--1415 (2008; Zbl 1132.26006)], where the relevance of a ``deeper theory of representations'' to the Brascam-Lieb inequality in harmonic analysis and its theory is hinted at. Here, they also give a concise overview of notation and their results.
The second section defines the BL operator \(T_{V,\sigma}\) associated to real \(Q\)-representation \(V\) and weight \(\sigma\) (inspired by [\textit{A. Garg} et al., Geom. Funct. Anal. 28, No. 1, 100--145 (2018; Zbl 1387.68133)]) and its capacity (Definition 4). The section then proves Theorem 1, using a criterion also from [loc. cit.].
Section 3 provides an explicit formula for the capacity of a quiver datum \((V, \sigma)\) (Lemma 8, Corollary 9) and defines the notion of BL constants \(BL_Q\) for quiver datum (Definition 10).
In Section 4, the notion of a geometric quiver datum is introduced (Definition 12). In the language of quiver invariant theory, generalising a result of [\textit{A. D. King}, Q. J. Math., Oxf. II. Ser. 45, No. 180, 515--530 (1994; Zbl 0837.16005)] and using a quiver version of a result of Kempf-Ness on closed orbits, the authors prove Theorem 2. This result includes a factorisation of the capacity of quiver data, the equivalence of existence of Gaussian extremals for \((V, \sigma)\) and \(V\) being \(\sigma\)-polystable, and that the uniqueness of Gaussian extremals implies that \(V\) is \(\sigma\)-stable.
Finally, in Section 5, the authors rephrase their main results in terms of BL constants and mention first applications [\textit{C. Chindris} and \textit{D. Kline}, J. Algebra 577, 210--236 (2021; Zbl 1467.16014)] and [\textit{C. Chindris} and \textit{D. Kline}, J. Pure Appl. Algebra 227, No. 3, Article ID 107234 (2023; Zbl 1517.16012)].
Reviewer: Sebastian Eckert (Bielefeld)Decomposition of topological Azumaya algebrashttps://zbmath.org/1521.160242023-11-13T18:48:18.785376Z"Arcila-Maya, Niny"https://zbmath.org/authors/?q=ai:arcila-maya.ninyBrauer groups of fields.
Classically, given a field \(k\), we can define the \textit{Brauer group} \(\mathrm{Br}(k)\) to be the (torsion) abelian group whose elements are equivalence classes of central simple algebras over \(k\), under the equivalence relation
\[
A\sim B \text{ if and only if } \text{ there exist } m, n \geq 1 \text{ such that } M_m(A) \cong M_n(B),
\]
and with group operation (usually written \(+\)) induced by the tensor product operation \(\otimes_k\).
If \(D\) is a division algebra over \(k\) with \(\deg(D) = d\), and \(d = ef\) for two coprime integers \(e\) and \(f\), then it is known that there exist division algebras \(E\) and \(F\), with \(\deg(E) = e\) and \(\deg(F) = f\), such that \(D = E\otimes_k F\).
The Artin-Wedderburn theorem implies that every class \(\alpha\in \mathrm{Br}(k)\) contains a unique division algebra \(D\), and the \textit{index} of \(\alpha\) is defined to be \(\mathrm{ind}(\alpha) := \deg(D) := \sqrt{\dim_k(D)}\) (which is always an integer) for this \(D\). This gives us a notion of \emph{prime decomposition} on the index of a Brauer class: if \(\mathrm{ind}(\alpha) = d\), and \(d = ef\) as above, then there exist classes \(\beta, \gamma\in \mathrm{Br}(k)\) with \(\mathrm{ind}(\beta) = e\) and \(\mathrm{ind}(\gamma) = f\) such that \(\alpha = \beta + \gamma\).
Brauer groups of Azumaya algebras.
For the remainder of the review, \(X\) will be a finite-dimensional CW complex.
A \textit{topological Azumaya algebra of degree \(n\)} over \(X\) is a bundle of complex algebras that is locally isomorphic to \(M_n(\mathbb{C})\). Topological Azumaya algebras also admit a notion of tensor product (just perform the tensor product of complex algebras on fibres), and so we can define an equivalence relation on topological Azumaya algebras
\[
\mathcal{A} \sim \mathcal{B} \text{ if and only if } \text{ there exist vector bundles } \mathcal{E}, \mathcal{F} \text{ such that } \mathcal{A}\otimes \mathrm{End}(\mathcal{E}) \cong \mathcal{B}\otimes \mathrm{End}(\mathcal{F}),
\]
from which we may define the \emph{Brauer group} \(\mathrm{Br}(X)\) as before.
A theorem of Serre shows that, in this setting, \(\mathrm{Br}(X)\) may be realised cohomologically as \(H^3(X,\mathbb{Z})_{\mathrm{tors}}\), which raises questions such as the following. Given \(\alpha\in H^3(X,\mathbb{Z})_{\mathrm{tors}}\), and interpreting it as a Brauer class: for which integers \(n\) does there exist a topological Azumaya algebra \(\mathcal{A}\in \alpha\)?
Define \(\mathrm{ind}(\alpha)\) to be the greatest common divisor of all such \(n\). This is an appropriate analogue for the definition for fields, and we may ask how well-behaved it is. For instance, there a notion of prime decomposition in this setting?
Antieau and Williams answered this question in the negative. For every odd integer \(n > 1\), they constructed a CW complex \(X\) with \(\dim(X) = 6\) and a topological Azumaya algebra \(\mathcal{A}\) with \(\deg(\mathcal{A}) = 2n\) with the following property: \(\mathcal{A}\) admits no decomposition \(\mathcal{A} \cong \mathcal{B} \otimes \mathcal{C}\) of topological Azumaya algebras where \(\deg(\mathcal{B}) = 2\) and \(\deg(\mathcal{C}) = n\).
The main result of the current paper is a positive result:
Theorem. Let \(m\) and \(n\) be coprime positive integers with \(m < n\). Suppose \(\dim(X) \leq 2m+1\). If \(\deg(\mathcal{A}) = mn\), then there exist \(\mathcal{B}\) and \(\mathcal{C}\) with \(\deg(\mathcal{B}) = m\) and \(\deg(\mathcal{C}) = n\) such that \(\mathcal{A} = \mathcal{B} \otimes \mathcal{C}\).
This result states that there \textit{is} in fact a limited notion of prime decomposition, as long as the dimension is sufficiently small compared to one of the coprime factors. The author also discusses the cohomological obstruction to uniqueness of \(\mathcal{B}\) and \(\mathcal{C}\). Finally, the author outlines a construction of an Azumaya algebra of dimension \(2m+2\) which cannot be decomposed, presumably showing the strictness of this bound. (However, this construction relies on the existence of a generator for \(\pi_{2m+2}(BPU_{mn})\), the \((2m+2)\)th homotopy group of the classifying space for the projective unitary group of order \(mn\): due to my own lack of familiarity with this group, I am unable to say for which values of \(m\) and \(n\) this exists.)
Reviewer: William Woods (Essex)Anticommutative algebras of the third levelhttps://zbmath.org/1521.170012023-11-13T18:48:18.785376Z"Volkov, Yury"https://zbmath.org/authors/?q=ai:volkov.yu-sIn the context of the paper, the level of a (generally, nonassociative algebra) is, roughly, in how many steps the algebra can be degenerated to the algebra with trivial multiplication. The author continues his previous investigations and describes finite-dimensional anticommutative algebras of level \(3\), and singles among them Lie algebras. The Inönü-Wigner contractions (that is, when the algebra is split into the direct sum of a subalgebra and a vector subspace, and the multiplication on the vector subspace becomes trivial under degeneration) play a role.
The paper is very technical, with a lot of computational details.
Reviewer: Pasha Zusmanovich (Ostrava)On some subspaces of the exterior algebra of a simple Lie algebrahttps://zbmath.org/1521.170172023-11-13T18:48:18.785376Z"Charbonnel, Jean-Yves"https://zbmath.org/authors/?q=ai:charbonnel.jean-yvesSummary: In this article, we are interested in some subspaces of the exterior algebra of a simple Lie algebra \(\mathfrak{g} \). In particular, we prove that some graded subspaces of degree \(d\) generate the \({\mathfrak{g}} \)-module \(\bigwedge^d{\mathfrak{g}}\) for some integers \(d\).Categorification via blocks of modular representations for \(\mathfrak{sl}_n\)https://zbmath.org/1521.170372023-11-13T18:48:18.785376Z"Nandakumar, Vinoth"https://zbmath.org/authors/?q=ai:nandakumar.vinoth"Zhao, Gufang"https://zbmath.org/authors/?q=ai:zhao.gufangSummary: Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of \(\mathfrak{sl}_2\), where they use singular blocks of category \(\mathcal{O}\) for \(\mathfrak{sl}_n\) and translation functors. Here we construct a positive characteristic analogue using blocks of representations of \(\mathfrak{s}\mathfrak{l}_n\) over a field \(\mathbf{k}\) of characteristic \(p\) with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmannians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmannians. This is part of a larger project to give a combinatorial approach to Lusztig's conjectures for representations of Lie algebras in positive characteristic.Local opers with two singularities: the case of \(\mathfrak{sl}(2)\)https://zbmath.org/1521.170522023-11-13T18:48:18.785376Z"Fortuna, Giorgia"https://zbmath.org/authors/?q=ai:fortuna.giorgia"Lombardo, Davide"https://zbmath.org/authors/?q=ai:lombardo.davide-m"Maffei, Andrea"https://zbmath.org/authors/?q=ai:maffei.andrea"Melani, Valerio"https://zbmath.org/authors/?q=ai:melani.valerioSummary: We study local opers with two singularities for the case of the Lie algebra \(\mathfrak{sl}(2)\), and discuss their connection with a two-variables extension of the affine Lie algebra. We prove an analogue of the Feigin-Frenkel theorem describing the centre at the critical level, and an analogue of a result by Frenkel and Gaitsgory that characterises the endomorphism rings of Weyl modules in terms of functions on the space of opers.The Poisson spectrum of the symmetric algebra of the Virasoro algebrahttps://zbmath.org/1521.170632023-11-13T18:48:18.785376Z"Sierra, Susan J."https://zbmath.org/authors/?q=ai:sierra.susan-j"Petukhov, Alexey V."https://zbmath.org/authors/?q=ai:petukhov.alexey-vSummary: Let \(W = \mathbb{C}[t,t^{-1}]\partial_t\) be the \textit{Witt algebra} of algebraic vector fields on \(\mathbb{C}^\times\) and let \(Vir\) be the \textit{Virasoro algebra}, the unique nontrivial central extension of \(W\). In this paper, we study the Poisson ideal structure of the symmetric algebras of \(Vir\) and \(W\), as well as several related Lie algebras. We classify prime Poisson ideals and Poisson primitive ideals of \(\operatorname{S}(Vir)\) and \(\operatorname{S}(W)\). In particular, we show that the only functions in \(W^*\) which vanish on a nontrivial Poisson ideal (that is, the only maximal ideals of \(\operatorname{S}(W)\) with a nontrivial Poisson core) are given by linear combinations of derivatives at a finite set of points; we call such functions \textit{local}. Given a local function \(\chi \in W^*\), we construct the associated Poisson primitive ideal through computing the algebraic symplectic leaf of \(\chi \), which gives a notion of coadjoint orbit in our setting. As an application, we prove a structure theorem for subalgebras of \(Vir\) of finite codimension and show, in particular, that any such subalgebra of \(Vir\) contains the central element \(z\), substantially generalising a result of Ondrus and Wiesner on subalgebras of codimension one. As a consequence, we deduce that \(\operatorname{S}(Vir)/(z-\zeta )\) is Poisson simple if and only if \(\zeta \neq ~0\).Tamarkin-Tsygan calculus and chiral Poisson cohomologyhttps://zbmath.org/1521.170702023-11-13T18:48:18.785376Z"Bouaziz, Emile"https://zbmath.org/authors/?q=ai:bouaziz.emileSummary: We construct and study some vertex theoretic invariants associated with Poisson varieties, specializing in the conformal weight \(0\) case to the familiar package of Poisson homology and cohomology. In order to do this conceptually, we sketch a version of the \textit{calculus}, in the sense of [\textit{D. Tamarkin} and \textit{B. Tsygan}, Methods Funct. Anal. Topol. 6, No. 2, 85--100 (2000; Zbl 0965.58010)], adapted to the context of vertex algebras. We obtain the standard theorems of Poisson (co)homology in this \textit{chiral} context. This is part of a larger project related to promoting noncommutative geometric structures to chiral versions of such.Universal cohomology theorieshttps://zbmath.org/1521.180052023-11-13T18:48:18.785376Z"Barbieri-Viale, Luca"https://zbmath.org/authors/?q=ai:barbieri-viale.lucaThe axiomatic approach to homology theories, in particular to singular versus cellular homology, as it was introduced by \textit{S. Eilenberg} and \textit{N. E. Steenrod} [Proc. Natl. Acad. Sci. USA 31, 117--120 (1945; Zbl 0061.40504); Foundations of algebraic topology. Princeton, NJ: Princeton University Press (1952; Zbl 0047.41402)] in topology, has been largely influential and their unicity theorem quite astonishing. The first key step in this story was taken around the years 1945--1952, while a ramified study of topological generalized (co)homology theories emerged for a start [\textit{E. Dyer}, Cohomology theories. Reading, MA: The Benjamin/Cummings Publishing Company (1969; Zbl 0182.57002)].
A parallel history can be seen in Grothendieck construction of a Weil cohomology in algebraic geometry, which was started from a wish-list of axioms and was afforded in the years 1958--1966 after two other key foundational steps, the first step being in homological algebra with the concept of satellite and that of \(\partial\)-functor [\textit{H. Cartan} and \textit{S. Eilenberg}, Homological algebra. Princeton, NJ: Princeton University Press (1956; Zbl 0075.24305); \textit{A. Grothendieck}, Tôhoku Math. J. (2) 9, 119--221 (1957; Zbl 0118.26104)] while the second step being an introduction of Grothendieck topologies. Notably, the stride from Weil cohomology to Grothendieck motives and motivic cohomology was originally based on the third foundational step of the Tannakian formalism, which is still dependent on the standard conjectures [\textit{P. Deligne} and \textit{J. S. Milne}, Lect. Notes Math. 900, 101--228 (1982; Zbl 0477.14004)]. Broadly speaking, the category of motives in Grothendieck vision can be regarded as a way to express a sort of abelian envelope of algebraic varieties, while motivic cohomology can be the abelian avatar of a variety.
\textit{P. Freyd} [Proc. Conf. Categor. Algebra, La Jolla 1965, 95--120 (1966; Zbl 0202.32402)] considered the question how nicely a given category can be represented in an abelian category, though his universal abelian category of an additive category has not been linked to the construction of motives. Freyd also observed that there is an embedding of a triangulated category in an abelian category which is universal with respect to homological functors, while \textit{A. Heller} [Bull. Am. Math. Soc. 74, 28--63 (1968; Zbl 0177.25605)] constructed a universal homology in a stable abelian category.
Actually, on the algebraic side of the story, around 1997, Nori provided a universal abelian category, making use of a variant of the Tannakian formalism [\textit{A. Huber} and \textit{S. Müller-Stach}, Periods and Nori motives. Cham: Springer (2017; Zbl 1369.14001)], while \textit{P. Deligne} [in: Galois groups over \(\mathbb{Q}\), Proc. Workshop, Berkeley/CA (USA) 1987, Publ., Math. Sci. Res. Inst. 16, . 79--297 (1989; Zbl 0742.14022)] introduced the abelian category of mixed realizations and \textit{Y. André} [Publ. Math., Inst. Hautes Étud. Sci. 83, 5--49 (1996; Zbl 0874.14010)] proposed motivated cycles showing how to shirk the standard conjectures. Nontheless, Nori's idea as well as André's and Deligne's is based on the Tannakian formalism, making use of existing fiber functors, so that the standard conjectures remain unsolved. The approach due to \textit{V. Voevodsky} [Sel. Math., New Ser. 2, No. 1, 111--153 (1996; Zbl 0871.14016)] and \textit{J. Ayoub} [J. Reine Angew. Math. 693, 1--149 (2014; Zbl 1299.14020)] gives the construction of a triangulated category of motives in place of the abelian category, though the correct motivic t-structure is missing.
The unified (co)homological framework in [\textit{L. Barbieri-Viale}, J. Pure Appl. Algebra 221, No. 7, 1565--1588 (2017; Zbl 1375.14019)] is also a first step toward the construction of motives independently of fiber functors, being settled in the language of categorical model theory. Its translation in the language of representations of quivers started in parallel with [\textit{L. Barbieri-Viale} and \textit{M. Prest}, Rend. Semin. Mat. Univ. Padova 139, 205--224 (2018; Zbl 1395.18012)]. Actually, Freyd's universal abelian category is linked with the construction of Nori motives as well as with the triangulated categories of Voevodsky motives. A tensor version of Freyd's universal abelian category provides tensor product of motives, e.g. for abelian categories modeles on a given cohomology satisfying Künnetuh formula [\textit{L. Barbieri-Viale} et al., Pac. J. Math. 306, No. 1, 1--30 (2020; Zbl 1454.14062); \textit{L. Barbieri-Viale} and \textit{M. Prest}, J. Pure Appl. Algebra 224, No. 6, Article ID 106267, 13 p. (2020; Zbl 1439.14077)].
The principal objective in this paper is to show that there is a simple algebraic picture providing universal (co)homology theories in abelian categories, independent of the Tannakian formalism, revisiting and developing the previously hinted construction of theoretical motives [\textit{L. Barbieri-Viale}, J. Pure Appl. Algebra 221, No. 7, 1565--1588 (2017; Zbl 1375.14019)]. This unified framework for (co)homology theories on any fixed category \(\mathcal{C}\) with values in variable abelian categories \(\mathcal{A}\) is achieved through the solution of representability problems.
Reviewer: Hirokazu Nishimura (Tsukuba)The stability manifold of local orbifold elliptic quotientshttps://zbmath.org/1521.180112023-11-13T18:48:18.785376Z"Rota, Franco"https://zbmath.org/authors/?q=ai:rota.francoSummary: We study the stability manifold of local models of orbifold quotients of elliptic curves. In particular, we show that a region of the stability manifold is a covering space of the regular set of the Tits cone of the associated elliptic root system. The construction requires an explicit description of the McKay correspondence [\textit{T. Bridgeland} et al., J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] for \(A_N\) surface singularities and a study of wall-crossing phenomena.Complete \(L_\infty \)-algebras and their homotopy theoryhttps://zbmath.org/1521.180212023-11-13T18:48:18.785376Z"Rogers, Christopher L."https://zbmath.org/authors/?q=ai:rogers.christopher-lIn previous joint work [J. Algebra 430, 260--302 (2015; Zbl 1327.17019)] with \textit{V. Dolgushev}, the author generalized a result of E. Getzler [Zbl 1246.17025], establishing a connection between the homotopy theory of complete \(L_{\infty}\)-algebras and Kan complexes, which is a generalization of the classical theorem of \textit{W. M. Goldman} and \textit{J. J. Millson} [Publ. Math., Inst. Hautes Étud. Sci. 67, 43--96 (1988; Zbl 0678.53059), Theorem 4.8] and which has turned out to be useful in a variety of applications beyond deformation theory, including the rational homotopy theory of mapping spaces [\textit{A. Berglund}, Homology Homotopy Appl. 17, No. 2, 343--369 (2015; Zbl 1347.55010); \textit{B. Fresse} and \textit{T. Willwacher}, ``Mapping spaces for DG Hopf cooperads and homotopy automorphisms of the rationalization of $E_n$-operads'', Preprint, \url{arXiv:2003.02939}] and operadic homotopical algebras [\textit{V. A. Dolgushev} et al., Adv. Math. 274, 562--605 (2015; Zbl 1375.18053); \textit{D. Robert-Nicoud}, Algebr. Geom. Topol. 19, No. 3, 1453--1476 (2019; Zbl 1475.17032)].
This paper further develops and strengthens the above result. The author first gives a finer-grained description of the homotopy theory within the category \(\widehat{\mathsf{Lie}}_{\infty}\), where \ is the category of complete filtered \(L_{\infty}\)-algebras and filtration-compatible weak \(L_{\infty}\)-morphisms. He then extends the aforementioned ``\(L_{\infty}\) Goldman-Millson Theorem'' by establishing the compatibility of this additional homotopical structure with the Kan-Quillen model structure for simpllcial sets. Finally, he uses this machinery to build an obstruction theory for the general problem of lifting Maurer-Cartan elements through an arbitrary \(\infty\)-morphism.
Reviewer: Hirokazu Nishimura (Tsukuba)Finite simple groups acting with fixity 3 and their occurrence as groups of automorphisms of Riemann surfaceshttps://zbmath.org/1521.200042023-11-13T18:48:18.785376Z"Salfeld, Patrick"https://zbmath.org/authors/?q=ai:salfeld.patrick"Waldecker, Rebecca"https://zbmath.org/authors/?q=ai:waldecker.rebeccaSummary: The results in this article are based on the classification of finite simple groups that act with fixity 3. Motivated by the theory of Riemann surfaces, we investigate which ones of these groups act faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most three fixed points on each non-regular orbit and at most four fixed points in total. In each case, we give detailed information about the possible branching data of the surface.Eulerian representations for real reflection groupshttps://zbmath.org/1521.200792023-11-13T18:48:18.785376Z"Brauner, Sarah"https://zbmath.org/authors/?q=ai:brauner.sarahSummary: The Eulerian idempotents, first introduced for the symmetric group and later extended to all reflection groups, generate a family of representations called the Eulerian representations that decompose the regular representation. In Type \(A\), the Eulerian representations have many elegant but mysterious connections to rings naturally associated with the braid arrangement. Here, we unify these results and show that they hold for any reflection group of coincidental type -- that is, \(S_n\), \(B_n\), \(H_3\) or the dihedral group \(I_2(m)\) -- by giving six characterizations of the Eulerian representations, including as components of the associated graded Varchenko-Gelfand ring \(\mathcal{V}\). As a consequence, we show that Solomon's descent algebra contains a commutative subalgebra generated by sums of elements with a fixed number of descents if and only if \(W\) is coincidental. More generally, for any finite real reflection group, we give a case-free construction of a family of Eulerian representations described by a flat decomposition of the ring \(\mathcal{V}\).Integral exotic sheaves and the modular Lusztig-Vogan bijectionhttps://zbmath.org/1521.201032023-11-13T18:48:18.785376Z"Achar, Pramod N."https://zbmath.org/authors/?q=ai:achar.pramod-n"Hardesty, William"https://zbmath.org/authors/?q=ai:hardesty.william-d"Riche, Simon"https://zbmath.org/authors/?q=ai:riche.simonSummary: Let \(G\) be a reductive algebraic group over an algebraically closed field \(\Bbbk\) of pretty good characteristic. The Lusztig-Vogan bijection is a bijection between the set of dominant weights for \(G\) and the set of irreducible \(G\)-equivariant vector bundles on nilpotent orbits, conjectured by \textit{G. Lusztig} [J. Fac. Sci., Univ. Tokyo, Sect. I A 36, No. 2, 297--328 (1989; Zbl 0688.20020)] and \textit{D. A. Vogan jun.} [in: Representation theory of Lie groups. Lecture notes from the Graduate summer school program, Park City, UT, USA, July 13--31, 1998. Providence, RI: American Mathematical Society (AMS). 177--238 (2000; Zbl 0948.22013)] independently, and constructed in full generality by \textit{R. Bezrukavnikov} [Represent. Theory 7, 1--18 (2003; Zbl 1065.20055)]. In characteristic 0, this bijection is related to the theory of 2-sided cells in the affine Weyl group, and plays a key role in the proof of the Humphreys conjecture on support varieties of tilting modules for quantum groups at a root of unity. In this paper, we prove that the Lusztig-Vogan bijection is (in a way made precise in the body of the paper) independent of the characteristic of \(\Bbbk \). This allows us to extend all of its known properties from the characteristic-0 setting to the general case. We also expect this result to be a step towards a proof of the Humphreys conjecture on support varieties of tilting modules for reductive groups in positive characteristic.Peterzil-Steinhorn subgroups and \(\mu\)-stabilizers in ACFhttps://zbmath.org/1521.201082023-11-13T18:48:18.785376Z"Kamensky, Moshe"https://zbmath.org/authors/?q=ai:kamensky.moshe"Starchenko, Sergei"https://zbmath.org/authors/?q=ai:starchenko.sergei"Ye, Jinhe"https://zbmath.org/authors/?q=ai:ye.jinheSummary: We consider \(G\), a linear algebraic group defined over \(\Bbbk\), an algebraically closed field (ACF). By considering \(\Bbbk\) as an embedded residue field of an algebraically closed valued field \(K\), we can associate to it a compact \(G\)-space \(S^\mu_G(\Bbbk)\) consisting of \(\mu\)-types on \(G\). We show that for each \(p_\mu \in S^\mu_G(\Bbbk)\), \(\mathrm{Stab}^\mu (p) = \mathrm{Stab}(p_\mu)\) is a solvable infinite algebraic group when \(p_\mu\) is centered at infinity and residually algebraic. Moreover, we give a description of the dimension of \(\mathrm{Stab}(p_\mu)\) in terms of the dimension of \(p\).Topological Frobenius reciprocity and invariant Hermitian formshttps://zbmath.org/1521.220092023-11-13T18:48:18.785376Z"Bratten, Tim"https://zbmath.org/authors/?q=ai:bratten.tim"Natale, Mauro"https://zbmath.org/authors/?q=ai:natale.mauroLet \(G\) a complex connected reductive group. \(G_0\) a real for for \(G\). \(P\) a parabolic subgroup of \(G\) so that it admits a Levi factor \(L\) with the property \(G_0 \cap P=G_0 \cap L\). \(K\) is the complexification of a maximal compact subgroup for \(G_0\). We denote by \(M_{\min}, M_{\max}\) Schmid's minimal and maximal globalizations of a \((\mathfrak g, K)\)-module. Next, we describe a topological Frobenius reciprocity shown by the authors. Let \(\mathfrak u\) be the nilradical of the Lie algebra of \(P\) and suppose \(M_{\max}\) is a quasisimple maximal globalization for a Harish-Chandra module for \(G_{0}\). Let \(H_{q}( \mathfrak u,M_{\max})_\sigma\) denote the corresponding \(Z(\mathfrak l)\) eigenspace corresponding to the character \(\sigma\). Then the right adjoint to the functor of geometric induction is given by \(M_{\mathrm{glob}} \rightarrow H_{q}( \mathfrak u,M_{\max})_\sigma \) where \(M_{\mathrm{glob}}\) is a quasisimple finite length admissible representation for \(G_{0}\). Under suitable hypothesis the authors show:
\[
\Hom_{ G_{0}}(H_c^q(S,\mathcal O(P,V_{\min})), M_{\mathrm{glob}})\cong \Hom_{L_0}(V_{\min}, H_{q}( \mathfrak u,M_{\max}))
\]
for every quasisimple finite length admissible representation \(M_{\mathrm{glob}}\). Dualizing, letting \(s\) be the complex dimension of \(Q\) and giving a completely formal definition to the representation \(H^s(S ,\mathcal O(\mathfrak p, W_{\max})))\) they obtain the natural isomorphism
\[
\Hom_{ G_{0}} (M_{\min}, H^s(\mathfrak p, \mathcal O(S, W_{\max}))\cong \Hom_{L_0}(H_s(\mathfrak u,M_{\min}),W_{\max}))
\]
which is valid at least when \(M_{\min}\) is quasisimple and \(W_{\max}\) has an infinitesimal character that satisfies an appropriate regular dominant condition. Their proof of the formula is based on a geometric construction of \(H_c^q(S,\mathcal O(P,V_{\min})\).
Reviewer: Jorge Vargas (Córdoba)Some explicit calculations of cohomology groups of \((\varphi,\Gamma)\)-moduleshttps://zbmath.org/1521.220192023-11-13T18:48:18.785376Z"Gaisin, Ildar"https://zbmath.org/authors/?q=ai:gaisin.ildarSummary: This is a survey article that advertises the idea of a \(p\)-adic Langlands correspondence for \(\mathrm{GL}_2(\mathbf{Q}_p)\) in the setting of not necessarily étale \((\varphi,\Gamma)\)-modules of rank 2. In particular the cohomology of various (semi)-groups is considered.Quadrangular \(\mathbb{Z}_p^l\)-actions on Riemann surfaceshttps://zbmath.org/1521.300512023-11-13T18:48:18.785376Z"Hidalgo, Ruben A."https://zbmath.org/authors/?q=ai:hidalgo.ruben-antonioSummary: Let \(p \geqslant 3\) be a prime integer and, for \(l \geqslant 1\), let \(G \cong\mathbb{Z}_p^l\) be a group of conformal automorphisms of some closed Riemann surface \(S\) of genus \(g \geqslant 2\). By the Riemann-Hurwitz formula, either \(p \leqslant g+1\) or \(p=2\,g+1\). If \(l=1\) and \(p=2\,g+1\), then \(S/G\) is the sphere with exactly three cone points and, if moreover \(p \geqslant 11\), then \(G\) is the unique \(p\)-Sylow subgroup of \(\mathrm{Aut}(S)\). If \(l=1\) and \(p=g+1\), then \(S/G\) is the sphere with exactly four cone points and, if moreover \(p \geqslant 7\), then \(G\) is again the unique \(p\)-Sylow subgroup. The above unique facts permitted many authors to obtain algebraic models and the corresponding groups \(\mathrm{Aut}(S)\) in these situations. Now, let us assume \(l \geqslant 2\). If \(p \geqslant 5\), then either (1) \(p^l \leqslant g-1\) or (2) \(S/G\) has genus zero, \(p^{l-1}(p-3) \leqslant 2(g-1)\) and \(2 \leqslant l \leqslant r-1\), where \(r \geqslant 3\) is the number of cone points of \(S/G\). Let us assume we are in case (2). If \(r=3\), then \(l=2\) and \(S\) happens to be the classical Fermat curve of degree \(p\), whose group of automorphisms is well known. The next case, \(r=4\), is studied in this paper. We provide an algebraic curve representation for \(S\), a description of its group of conformal automorphisms, a discussion of its field of moduli and an isogenous decomposition of its Jacobian variety.The Weil-Petersson current on Douady spaceshttps://zbmath.org/1521.320102023-11-13T18:48:18.785376Z"Axelsson, Reynir"https://zbmath.org/authors/?q=ai:axelsson.reynir"Schumacher, Georg"https://zbmath.org/authors/?q=ai:schumacher.georgSummary: The Douady space of compact subvarieties of a Kähler manifold is equipped with the Weil-Petersson current, which is everywhere positive with local continuous potentials, and of class \(C^\infty\) when restricted to the locus of smooth fibers. There a Quillen metric is known to exist, whose Chern form is equal to the Weil-Petersson form. In the algebraic case, we show that the Quillen metric can be extended to the determinant line bundle as a singular hermitian metric. On the other hand the determinant line bundle can be extended in such a way that the Quillen metric yields a singular hermitian metric whose Chern form is equal to the Weil-Petersson current. We show a general theorem comparing holomorphic line bundles equipped with singular hermitian metrics which are isomorphic over the complement of a snc divisor \(B\). They differ by a line bundle arising from the divisor and a flat line bundle. The Chern forms differ by a current of integration with support in \(B\) and a further current related to its normal bundle. The latter current is equal to zero in the case of Douady spaces due to a theorem of Yoshikawa on Quillen metrics for singular families over curves.
{{\copyright} 2021 The Authors. \textit{Mathematische Nachrichten} published by Wiley-VCH GmbH}Equivariant Oka theory: survey of recent progresshttps://zbmath.org/1521.320202023-11-13T18:48:18.785376Z"Kutzschebauch, Frank"https://zbmath.org/authors/?q=ai:kutzschebauch.frank"Lárusson, Finnur"https://zbmath.org/authors/?q=ai:larusson.finnur"Schwarz, Gerald W."https://zbmath.org/authors/?q=ai:schwarz.gerald-wIn this survey article, the authors present the development of equivariant Oka theory since 2015, to which they made the essential contributions of the last decade.
While ``plain'' Oka theory deals with questions where holomorphic geometric problems only have topological obstructions, equivariant Oka theory requires in addition that all the involved maps are equivariant w.r.t.\ a given group action.
To illustrate the flavour of the results, we only cite their Theorem A here:
Let \(G\) be a reductive complex Lie group acting holomorphically on Stein manifolds \(X\) and \(Y\). Consider the categorical quotients \(Q_X = X /\!/ G\) and \(Q_Y = Y /\!/ G\). Assume that \(Q := Q_X = Q_Y\). A \(G\)-diffeomorphism \(\Phi \colon X \to Y\) is called \emph{strict} if it induces a \(G\)-biholomorphism of the reduced fibres \((X_q)_{\mathrm{red}}\) with \((Y_q)_{\mathrm{red}}\) for all \(q \in Q\).
Theorem A. Let \(G\) be a reductive complex Lie group. Let \(X\) and \(Y\) be Stein \(G\)-manifolds with common quotient \(Q\). Suppose that there exists a strict \(G\)-diffeomorphism \(\Phi \colon X \to Y\). Then \(\Phi\) is homotopic, through strict \(G\)-diffeomorphisms, to a \(G\)-biholomorphism from \(X\) to \(Y\).
Further aspects of equivariant Oka theory presented in this article are: Applications to the linearisation problem; A parametric Oka principle for sections of a bundle \(E\) of homogeneous spaces for a group bundle \(G\); Application to the classification of generalised principal bundles with a group action; An equivariant version of Gromov's Oka principle based on a notion of a \(G\)-manifold being \(G\)-Oka. Finally, some open problems are presented.
Reviewer: Rafael B. Andrist (Ljubljana)Holomorphic isometries into homogeneous bounded domainshttps://zbmath.org/1521.320212023-11-13T18:48:18.785376Z"Loi, Andrea"https://zbmath.org/authors/?q=ai:loi.andrea"Mossa, Roberto"https://zbmath.org/authors/?q=ai:mossa.robertoThe main result of the paper says that, if \((M,g)\) is a Kähler maniford which admits a holomorphic isometry into a homogeneous bounded domain \(\Omega \), then \(g\) is a Kähler-Einstein metric. Furthermore the manifold \((M,g)\) cannot admit a holomorphic isometry into a complex Euclidean space. This result extends previous ones by \textit{E. Calabi} [Ann. Math. (2) 58, 1--23 (1953; Zbl 0051.13103)], \textit{M. Umehara} [J. Math. Soc. Japan 40, No. 3, 519--539 (1988; Zbl 0651.53046)], and by the authors [Proc. Am. Math. Soc. 149, No. 11, 4931--4941 (2021; Zbl 1483.53077)]. Let \(\psi : M\to \Omega \) be a holomorphic isometry from the Kähler manifold \(M\) into a homogenous bounded domain \(\Omega \). Since \(\Omega \) is isomorphic to a homogeneous Siegel domain, one can consider a holomorphic isometry \(\varphi \) from \(M\) into a homogeneous Siegel domain \(\mathcal{D}\). By a result of \textit{J. Dorfmeister} [Trans. Am. Math. Soc. 288, 293--305 (1985; Zbl 0567.53021)] any Kähler metric on \(\mathcal{D}\) is equivalent to a Kähler metric which admits a potential function of the form
\[
\sum _{k=1}^r \gamma _kF_k(z),
\]
where the rational functions \(F_k\) are expressed by means of the power functions introduced by \textit{S. Gindink} [``Analysis on homogeneous domain'', Russ. Math. Surv. 19, 1--89 (1964)], and \(r\) is the rank of \(\mathcal{D}\). On the Kähler manifold \((M,g)\) one considers a special kind of local Kähler potential \(D_p(z)\) around the point \(p\). Such a local potential has been introduced by \textit{E. Calabi} [Ann. Math. (2) 58, 1--23 (1953; Zbl 0051.13103)] and called \textit{diastasis}. Then one proves that this potential should be of the form \(D_p(z)=D_{\varphi (p)}^{\Omega }\bigl(\varphi (z)\bigr)\), where \(D_v^{\Omega }(u)\) is a Kähler potential on \(\mathcal{D}\) of the form
\[
D_v^{\Omega }(u)=\sum _{k=1}^r \gamma _k \log \Bigl(\frac{F_k(u,\bar{u})F_k(v,\bar{v})}{F_k(v,\bar{v})F_k(v,\bar{u})}\Bigr).
\]
Then one proves that this form of the potential \(D_p(z)\) implies that \(g\) is an Einstein metric. Furthermore assuming that the Kähler manifold \((M,g)\) admits a holomorphic isometry into a complex Euclidean space, one gets a contradiction to the diastasis property of the function \(\exp \bigl(D_{\varphi (p)}^{\Omega }(\varphi (z)\bigr)\).
Reviewer: Jacques Faraut (Paris)Algebraicity of foliations on complex projective manifolds, applicationshttps://zbmath.org/1521.320232023-11-13T18:48:18.785376Z"Campana, Frédéric"https://zbmath.org/authors/?q=ai:campana.fredericSummary: This is an expository text, originally intended for the ANR 'Hodgefun' workshop, twice reported, organised at Florence, villa Finaly, by B. Klingler. We show that holomorphic foliations on complex projective manifolds have algebraic leaves under a certain positivity property: the 'non pseudoeffectivity' of their duals. This permits to construct certain rational fibrations with fibres either rationally connected, or with trivial canonical bundle, of central importance in birational geometry. A considerable extension of the range of applicability is due to the fact that this positivity is preserved by the tensor powers of the tangent bundle. The results presented here are extracted from [\textit{A. Moreno Cañadas} et al., Electron Res. Arch. 30, No. 2, 661--682 (2022; Zbl 1510.16012)], which is inspired by the former results [\textit{H. Chang} et al., Electron Res. Arch. 29, No. 3, 2457--2473 (2021; Zbl 1502.17016); \textit{D. Jumaniyozov} et al., Electron Res. Arch. 29, No. 6, 3909--3993 (2021; Zbl 1511.17007)]. In order to make things as simple as possible, we present here only the projective versions of these results, although most of them can be easily extended to the logarithmic or 'orbifold' context.A note on the Łojasiewicz exponent of non-degenerate isolated hypersurface singularitieshttps://zbmath.org/1521.320332023-11-13T18:48:18.785376Z"Brzostowski, Szymon"https://zbmath.org/authors/?q=ai:brzostowski.szymonThe main result of this note is the following: the Łojasiewicz exponents of two Kouchnirenko non-degenerate holomorphic functions \(f,g : (\mathbb C^n,0) \to (\mathbb C,0)\) having an isolated singularity at zero and the same Newton diagram are equal. The proof uses Tessier's result [\textit{B. Teissier}, Invent. Math. 40, 267--292 (1977; Zbl 0446.32002)] that the Łojasiewicz exponent is constant along certain deformations. The verification that the deformations used in the proof satisfy the required conditions is given in a wider context of formal power series over any algebraically closed field.
For the entire collection see [Zbl 1429.00039].
Reviewer: Armin Rainer (Wien)Entropy and affine actions for surface groupshttps://zbmath.org/1521.370282023-11-13T18:48:18.785376Z"Labourie, François"https://zbmath.org/authors/?q=ai:labourie.francoisSummary: We give a short and independent proof of a theorem of \textit{J. Danciger} and \textit{T. Zhang} [Geom. Funct. Anal. 29, No. 5, 1369--1439 (2019; Zbl 1427.57028)]: surface groups with Hitchin linear part cannot act properly on the affine space. The proof is fundamentally different and relies on ergodic methods.
{{\copyright} 2022 The Authors. \textit{Journal of Topology} is copyright {\copyright} London Mathematical Society.}Invariant curves for endomorphisms of \(\mathbb{P}^1 \times \mathbb{P}^1\)https://zbmath.org/1521.371072023-11-13T18:48:18.785376Z"Pakovich, Fedor"https://zbmath.org/authors/?q=ai:pakovich.fedorSummary: Let \(A_1, A_2 \in \mathbb{C}(z)\) be rational functions of degree at least two that are neither Lattès maps nor conjugate to \(z^{\pm n}\) or \(\pm T_n\). We describe invariant, periodic, and preperiodic algebraic curves for endomorphisms of \((\mathbb{P}^1 (\mathbb{C}))^2\) of the form \((z_1, z_2) \rightarrow (A_1(z_1), A_2(z_2))\). In particular, we show that if \(A \in \mathbb{C}(z)\) is not a ``generalized Lattès map'', then any \((A, A)\)-invariant curve has genus zero and can be parametrized by rational functions commuting with \(A\). As an application, for \(A\) defined over a subfield \(K\) of \(\mathbb{C}\) we give a criterion for a point of \((\mathbb{P}^1(K))^2\) to have a Zariski dense \((A, A)\)-orbit in terms of canonical heights, and deduce from this criterion a version of a conjecture of \textit{S.-W. Zhang} [Surv. Differ. Geom. 10, 381--430 (2006; Zbl 1207.37057)]
on the existence of rational points with Zariski dense forward orbits. We also prove a result about functional decompositions of iterates of rational functions, which implies in particular that there exist at most finitely many \((A_1, A_2)\)-invariant curves of any given bi-degree \((d_1, d_2)\).Ranks of linear matrix pencils separate simultaneous similarity orbitshttps://zbmath.org/1521.470272023-11-13T18:48:18.785376Z"Derksen, Harm"https://zbmath.org/authors/?q=ai:derksen.harm"Klep, Igor"https://zbmath.org/authors/?q=ai:klep.igor"Makam, Visu"https://zbmath.org/authors/?q=ai:makam.visu"Volčič, Jurij"https://zbmath.org/authors/?q=ai:volcic.jurijSummary: This paper solves the two-sided version and provides a counterexample to the general version of the 2003 conjecture by \textit{D. Hadwin} and \textit{D. R. Larson} [J. Funct. Anal. 199, No. 1, 210--227 (2003; Zbl 1026.46043)]. Consider evaluations of linear matrix pencils \(L = T_0 + x_1 T_1 + \cdots + x_m T_m\) on matrix tuples as \(L( X_1, \ldots, X_m) = I \otimes T_0 + X_1 \otimes T_1 + \cdots + X_m \otimes T_m\). It is shown that ranks of linear matrix pencils constitute a collection of separating invariants for simultaneous similarity of matrix tuples. That is, \(m\)-tuples \(A\) and \(B\) of \(n \times n\) matrices are simultaneously similar if and only if \(\operatorname{rk} L(A) = \operatorname{rk} L(B)\) for all linear matrix pencils \(L\) of size \(mn\). Variants of this property are also established for symplectic, orthogonal, unitary similarity, and for the left-right action of general linear groups. Furthermore, a polynomial time algorithm for orbit equivalence of matrix tuples under the left-right action of special linear groups is deduced.Realizability of some Böröczky arrangements over the rational numbershttps://zbmath.org/1521.520102023-11-13T18:48:18.785376Z"Janasz, Marek"https://zbmath.org/authors/?q=ai:janasz.marek"Lampa-Baczyńska, Magdalena"https://zbmath.org/authors/?q=ai:lampa-baczynska.magdalena"Wójcik, Daniel"https://zbmath.org/authors/?q=ai:wojcik.daniel-kThe authors study the parameter spaces for Böröczky arrangements \(B_n\) of \(n\) lines, with \(n<12\). Here, \(B_n\) is the configuration of \(n\) lines arranged according to the following construction: considering an \(2n\)-gon inscribed in a circle, and fixing one of the \(2n\) vertices, which will be denoted by \(Q_0\), then \(Q_\alpha\) is the point constructed by rotating \(Q_0\) around the center of a circle by some angle \(\alpha\). Next the following \(n\) lines are considered
\[
B_n=\left\{Q_\alpha Q_{\pi-2\alpha}, \text{ were }\alpha=\frac{2k\pi}{n} \text{ for } k=0,1,\dots,n-1\right\}.
\]
If \(\alpha\equiv (\pi-2\alpha) \mod 2\pi\), then \(Q_\alpha Q_{\pi-2\alpha}\) is the tangent to the circle at the point \(Q_\alpha\).
The aim of this paper is to \textit{complete the picture} for a number of lines between 3 and 11 in Böröczky arrangements, and to establish the realizability of these configurations over the rational numbers. The latter means, that there exists a configuration of that number of lines with the same incidences between the lines and the intersection points, such that all points have rational coordinates.
The interest in these configurations seems to have been recently renewed, due to its connection to the containment problem in Commutative Algebra (see [\textit{A. Czapliński} et al., Adv. Geom. 16, No. 1, 77--82 (2016; Zbl 1333.13005)] and [\textit{Ł. Farnik} et al., J. Algebr. Comb. 50, No. 1, 39--47 (2019; Zbl 1419.52021)]).
The authors mention the works [\textit{M. Lampa-Baczyńska} and \textit{J. Szpond}, Geom. Dedicata 188, 103--121 (2017; Zbl 1366.14048)], and [\textit{Ł. Farnik} et al., Int. J. Algebra Comput. 28, No. 7, 1231--1246 (2018; Zbl 1403.52012)], where the parameter spaces of certain Böröczky arrangements are considered, as their inspiration for the actual work. Indeed, in [Lampa-Baczyńska and Szpond, loc. cit.] it was shown that the \(B_{12}\) arrangement is realizable over the rational numbers.
The authors use an algorithm based on ideas of [\textit{B. Sturmfels}, J. Symb. Comput. 11, No. 5--6, 595--618 (1991; Zbl 0766.14043)], combined with methods established in [Lampa-Baczyńska and Szpond, loc. cit.], from which they can conclude that all arrangements \(B_n\), \(n\leq 10\), are realizable over the rationals. They obtain, further, that \(B_{11}\) is not realizable over the rationals. As mentioned above, a proof that \(B_{12}\) is realizable over the rational numbers can be found in [Lampa-Baczyńska and Szpond, loc. cit.]. Thus, \(B_{11}\) is the only Böröczky configuration, up to \(n = 12\), which is nonrealizable over the rational numbers. Other results on realizability of \(B_n\) are known in the literature for \(n\in\{ 13, 14,15,16,18,24\}\).
For the entire collection see [Zbl 1509.14002].
Reviewer: Eugenia Saorín Gómez (Bremen)De Rham cohomology and semi-slant submanifolds in metallic Riemannian manifoldshttps://zbmath.org/1521.530172023-11-13T18:48:18.785376Z"Gök, Mustafa"https://zbmath.org/authors/?q=ai:gok.mustafaSummary: In this paper, we deal with the de Rham cohomology of semi-slant submanifolds in metallic Riemannian manifolds. Some necessary conditions for a semi-slant submanifold of metallic Riemannian manifolds are given to define a well-defined canonical de Rham cohomology class. Also, the non-triviality of such a cohomology class is discussed. Finally, an example is constructed to illustrate the main idea of the paper.Homogeneous Einstein metrics and butterflieshttps://zbmath.org/1521.530372023-11-13T18:48:18.785376Z"Böhm, Christoph"https://zbmath.org/authors/?q=ai:bohm.christoph.1"Kerr, Megan M."https://zbmath.org/authors/?q=ai:kerr.megan-mA Riemannian manifold \((M, g)\) is called Einstein if it has constant Ricci tensor, that is if \(\mathrm{Ric}(g) =\lambda\cdot g\), \(\lambda\in \mathbb{R}\). The present work refers to the homogeneous setting, where a compact homogeneous space \((M, g)\) is a compact Riemannian manifold on which a compact Lie group \(G\) acts transitively by isometries. Then, it has a presentation \(M = G/H\), where \(H\) is a compact subgroup of \(G\). General existence results are difficult to obtain. However, in [\textit{C. Böhm}, J. Differ. Geom. 67, No. 1, 79--165 (2004; Zbl 1098.53039); \textit{C. Böhm} et al., Geom. Funct. Anal. 14, No. 4, 681--733 (2004; Zbl 1068.53029)], the authors reduced the existence problem to the study of Lie algebraic objects, such as simplicial complexes and graphs. On the other hand, \textit{M. M. Graev} [Trans. Mosc. Math. Soc. 2012, 1--28 (2012; Zbl 1278.53043); translation from Tr. Mosk. Mat. O.-va 73, No. 1, 1--35 (2012)], associated with a compact homogeneous space \(G/H\) the nerve \(X_{G/H}\), whose non-contractibility implies the existence of a \(G\)-invariant Einstein metric on \(G/H\). The nerve \(X_{G/H}\) is a compact semialgebraic set defined Lie-theoretically by intermediate subgroups.
The first goal of the present paper is to present a very detailed analysis of Graev's work. The second one is to give a shorter proof of the following first author's result in [J. Differ. Geom. 67, No. 1, 79--165 (2004; Zbl 1098.53039)]: if the simplicial complex \(\Delta _{G/H}\) of a compact homogeneous space with finite fundamental group and \(G, H\) connected, is not contractible, then \(G/H\) admits a \(G\)-invariant Einstein metric.
The paper also reviews some known classification results, and proposes some open problems about homogeneous Einstein metrics.
Reviewer: Andreas Arvanitoyeorgos (Pátra)Calabi-Yau metrics with conical singularities along line arrangementshttps://zbmath.org/1521.530502023-11-13T18:48:18.785376Z"de Borbon, Martin"https://zbmath.org/authors/?q=ai:de-borbon.martin"Spotti, Cristiano"https://zbmath.org/authors/?q=ai:spotti.cristianoConical Kähler-Einstein metrics with singularities along a divisor became relevant in Kähler geometry since the seminal work of \textit{X. Chen} et al. [J. Am. Math. Soc. 28, No. 1, 183--197 (2015; Zbl 1312.53096)]. It is natural to ask what happens when the divisor becomes singular, and this paper deals with one of the simplest of such cases, namely when the singularity occurs along a line arrangement in the 2-dimensional complex projective space, with conical angles prescribed by weight data, and the local behavior near the intersections of the divisors for modeled on polyhedral Kähler cone metrics. The main result is an existence result of a Calabi-Yau metric with the prescribed singularity, based on an adaption of Yau's continuity path, and some extra linear theory. As a partially conjectural application, the authors discuss a log version of the Bogomolov-Miyaoka-Yau inequality.
Reviewer: Yang Li (Cambridge, MA)Comparison of Poisson structures on moduli spaceshttps://zbmath.org/1521.530602023-11-13T18:48:18.785376Z"Biswas, Indranil"https://zbmath.org/authors/?q=ai:biswas.indranil"Bottacin, Francesco"https://zbmath.org/authors/?q=ai:bottacin.francesco"Gómez, Tomás L."https://zbmath.org/authors/?q=ai:gomez.tomas-lLet \(X\) be an irreducible smooth complex curve of genus \(g\). Consider a fixed algebraic line bundle \(N\) on \(X\). Let \(E\) be an algebraic vector bundle of rank \(r\) and degree \(\delta\), and \(\theta \in H^{0}(X, \mathcal{End}(E)\otimes N)\). A Hitchin pair \((E, \theta)\) is called stable if
\[
\deg F\cdot r < \delta\cdot \mathrm{rk} F
\]
for all nonzero proper subbundles \(F\subset E\) for which \(\theta(F) \subset F\otimes E\).
Let \(\mathcal{M}\) denote the moduli space of stable Hitchin pairs. It is known that \(\mathcal{M}\) has a nontrivial algebraic Poisson structure.
Let \(S\) be the smooth complex quasi-projective surface defined by the total space of the line bundle \(N\). Associated to \((E, \theta)\) there exist a subscheme \(Y_{(E, \theta)}\) and a coherent sheaf \(\mathcal{L}_{(E, \theta)}\). The pair \((Y_{(E, \theta)}, \mathcal{L}_{(E, \theta)})\) is known as the spectral datum of \((E, \theta)\). The spectral datum of a Hitchin pair produces an isomorphism \(\Phi:\mathcal{M}\to \mathcal{P}\) into the moduli space of stable sheaves of pure dimension \(1\) on \(S\).
A certain section \(\sigma\) of \(N\otimes K_{X}\) gives rise to a Poisson structure on the surface \(S\). Here \(K_{X}\) denotes as usual the canonical bundle of \(X\). An algebraic Poisson structure on \(\mathcal{P}\) is constructed with the Poisson structure on \(S\).
The main result of this paper is that the isomorphism \(\Phi\) sends the Poisson structure on \(\mathcal{M}\) to the Poisson structure on \(\mathcal{P}\). This generalizes previous results, for the case of the symplectic form of the moduli space of Higgs bundles, found in [\textit{I. Biswas} and \textit{A. Mukherjee}, Commun. Math. Phys. 221, No. 2, 293--304 (2001; Zbl 1066.14038); \textit{J. Harnad} and \textit{J. Hurtubise}, J. Math. Phys. 49, No. 6, 062903, 21 p. (2008; Zbl 1152.81465); \textit{J. C. Hurtubise} and \textit{M. Kjiri}, Commun. Math. Phys. 210, No. 2, 521--540 (2000; Zbl 0984.37090)].
Reviewer: Pablo Suárez-Serrato (Ciudad de México)The Kähler-Ricci flow with log canonical singularitieshttps://zbmath.org/1521.530752023-11-13T18:48:18.785376Z"Chau, Albert"https://zbmath.org/authors/?q=ai:chau.albert"Ge, Huabin"https://zbmath.org/authors/?q=ai:ge.huabin"Li, Ka-Fai"https://zbmath.org/authors/?q=ai:li.ka-fai"Shen, Liangming"https://zbmath.org/authors/?q=ai:shen.liangmingSingularities for the Kähler-Ricci flow may develop in finite time even when starting with a nonsingular variety. The authors establish the existence of the Kähler-Ricci flow in the case of \(\mathbb Q\)-factorial projective varieties with log canonical singularities. This generalizes some of the existence results of \textit{J. Song} and \textit{G. Tian} [Invent. Math. 207, No. 2, 519--595 (2017; Zbl 1440.53116)] in the case of projective varieties with klt singularities. They also prove that the normalized Kähler-Ricci flow converges to a Kähler-Einstein metric with negative Ricci curvature on semi-log canonical models in the sense of currents. Further, they show that the weak Kähler-Ricci flow can be uniquely extended through the divisorial contractions and flips on \(\mathbb Q\)-factorial projective varieties with log canonical singularities.
Reviewer: Ljudmila Kamenova (New York)Kähler-Ricci flow on rational homogeneous varietieshttps://zbmath.org/1521.530762023-11-13T18:48:18.785376Z"Correa, Eder M."https://zbmath.org/authors/?q=ai:correa.eder-mIn this paper, the Kähler-Ricci flow on varieties of the form \(G^{\mathbb{C}}/P\) (rational homogeneous varieties) is considered starting from a homogeneous metric. Analytically, the problem has an explicit linear solution \(\omega(t)=\omega_0-t \mathrm{Ric}(\omega_0)\). The goals however are to study the maximal time of existence, curvature estimates, the boundedness or lack thereof of the diameter, etc. Theorem A addresses these questions. In particular, the Myers' theorem combined with Theorem A shows that the diameter is bounded above, thus providing evidence for a well-known conjecture. If the initial Kähler class arises from a divisor, Corollary B provides various relationships between the differential-geometric and algebro-geometric aspects of the divisor. The proof hinges on an explicit basis of the Kähler cone arising from Lie theory (and results of \textit{H. Azad} and \textit{I. Biswas} [J. Algebra 269, No. 2, 480--491 (2003; Zbl 1042.53047)]).
Reviewer: Vamsi Pritham Pingali (Bangalore)Blown-up intersection cochains and Deligne's sheaveshttps://zbmath.org/1521.550062023-11-13T18:48:18.785376Z"Chataur, David"https://zbmath.org/authors/?q=ai:chataur.david"Saralegi-Aranguren, Martintxo"https://zbmath.org/authors/?q=ai:saralegi-aranguren.martintxo"Tanré, Daniel"https://zbmath.org/authors/?q=ai:tanre.danielThis paper is one in a series of papers studying the so-called blown-up intersection cochains. It builds on results of Poincaré duality for sheaf cohomology. In particular both the sheafification \(\boldsymbol N^*\) of the presheaf of singular cochains and the sheafification \(\boldsymbol C^*_{BM}\) of the presheaf of relative locally finite singular cochains also called Borel-Moore chains are acyclic resolutions of the constant sheaf with values in \(\mathbb R\). With taking hypercohomology they have the same homology. There is also a compact support version of this. Taking dual sheaves \(\mathbb D\) we get a pairing \(\mathbb D(\boldsymbol C^*_{BM}[n])\cong \boldsymbol N^*\) and \(\mathbb D(\mathbb R)=\mathbb R\).
If the space is singular, Poincaré duality is not satisfied anymore. However Verdier duality still makes sense if the space is stratified. Then the intersection complex \(\boldsymbol IC^*_{\bar p}\) introduced by \textit{M. Goresky} and \textit{R. MacPherson} [Topology 19, 135--165 (1980; Zbl 0448.55004)] is a complex of sheaves which is self-dual.
The authors study blown-up singular cochains \(\tilde N^*_{\bar p}\) introduced in previous papers. They provide a cap product between \(\tilde N^*_{\bar p}\) and a generalization of \(\mathfrak{C}^{\bar p}_*(X;R)\) of the singular chain complex introduced by \textit{H. C. King} [Topology Appl. 20, 149--160 (1985; Zbl 0568.55003)] which is a quasi-isomorphism for values in any ring. This extends a Poincaré duality theorem by \textit{G. Friedman} and \textit{J. E. McClure} [Adv. Math. 240, 383--426 (2013; Zbl 1280.55003)].
Reviewer: Elisa Hartmann (Göttingen)Transchromatic extensions in motivic bordismhttps://zbmath.org/1521.550082023-11-13T18:48:18.785376Z"Beaudry, Agnès"https://zbmath.org/authors/?q=ai:beaudry.agnes"Hill, Michael A."https://zbmath.org/authors/?q=ai:hill.michael-a"Shi, Xiaolin Danny"https://zbmath.org/authors/?q=ai:shi.xiaolin-danny"Zeng, Mingcong"https://zbmath.org/authors/?q=ai:zeng.mingcongThe paper under review considers multiplicative structure and the Toda brackets in the motivic bordism spectrum \(MGL\), the Real bordism spectrum \(MU_{\mathbb{R}}\), the motivic Morava \(K\)-theory spectra \(k_{GL}(n)\), and equivariant Morava \(K\)-theory spectra \(k_{\mathbb{R}}(n)\). A remarkable feature of these computations is the fact that these Toda brackets exhibit transchromatic phenomena. This demonstrates that killing certain classes in the homotopy of \(MGL\), for example, has a more complicated effect than one might naively guess. The arguments in the paper are elegant and the succinct exposition communicates the context and impact of the results well.
Reviewer: Gabriel Angelini-Knoll (Paris)Generalized stochastic areas, winding numbers, and hyperbolic Stiefel fibrationshttps://zbmath.org/1521.600472023-11-13T18:48:18.785376Z"Baudoin, Fabrice"https://zbmath.org/authors/?q=ai:baudoin.fabrice"Demni, Nizar"https://zbmath.org/authors/?q=ai:demni.nizar"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing.5This paper fit into a larger research project concerning the study of integrable functionals of Brownian motions on symmetric spaces.
More recently, Brownian winding functionals were studied in several papers in the complex projective space and the complex hyperbolic space. Here, the Brownian motion is studied on the non-compact Grassmann manifold \(\frac{U(n-k,k)}{U(n-k)U(k)}.\) Realizing the Brownian motion on \(HG_{n,k}\) as a matrix diffusion process has the advantage to make available all the tools from stochastic calculus and random matrix theory.
Reviewer: Rózsa Horváth-Bokor (Budakalász)Ulrich complexityhttps://zbmath.org/1521.680702023-11-13T18:48:18.785376Z"Bläser, Markus"https://zbmath.org/authors/?q=ai:blaser.markus"Eisenbud, David"https://zbmath.org/authors/?q=ai:eisenbud.david"Schreyer, Frank-Olaf"https://zbmath.org/authors/?q=ai:schreyer.frank-olafSummary: In this note we suggest a new measure of the complexity of polynomials, the Ulrich complexity. Valiant's conjecture on the exponential complexity of the permanent would imply exponential behavior of the Ulrich complexity as well, and this may be easier to prove. We compute some families of examples, one of which has provably exponential behavior.Liquid tensor experimenthttps://zbmath.org/1521.682452023-11-13T18:48:18.785376Z"Commelin, Johan"https://zbmath.org/authors/?q=ai:commelin.johan-mSummary (translated from the German): The Liquid Tensor Experiment is a project to verify the proof provided by Clausen and Scholze in 2019 of the ``main theorem of liquid vector spaces'' using computers. In the last few months, a team of mathematicians have largely completed this project. In this article we will explain, based on this developments, the potential of computer proof assistants for mathematical research.Computing circuit polynomials in the algebraic rigidity matroidhttps://zbmath.org/1521.682672023-11-13T18:48:18.785376Z"Malić, Goran"https://zbmath.org/authors/?q=ai:malic.goran"Streinu, Ileana"https://zbmath.org/authors/?q=ai:streinu.ileanaSummary: We present an algorithm for computing \textit{circuit polynomials} in the algebraic rigidity matroid \(\boldsymbol{\mathcal{A}}(\mathrm{CM}_n)\) associated to the Cayley-Menger ideal \(\mathrm{CM}_n\) for \(n\) points in 2D. It relies on combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in this ideal. We show that every rigidity circuit has a construction tree from \(K_4\) graphs based on this operation. Our algorithm performs an algebraic elimination guided by such a construction tree and uses classical resultants, factorization, and ideal membership. To highlight its effectiveness, we implemented the algorithm in Mathematica: it took less than 15 seconds on an example where a Gröbner basis calculation took 5 days and 6 hours. Additional speed-ups are obtained using non-\(K_4\) generators of the Cayley-Menger ideal and simple variations on our main algorithm.Quantum similarity index and Rényi complexity ratio of Kratzer type potential and compared with that of inverse square and Coulomb type potentialshttps://zbmath.org/1521.810202023-11-13T18:48:18.785376Z"Nath, Debraj"https://zbmath.org/authors/?q=ai:nath.debraj"Carbó-Dorca, Ramon"https://zbmath.org/authors/?q=ai:carbo-dorca.ramonSummary: In this paper, we define three sets of exact solutions for potentials of the type two-parameter Kratzer, inverse square, and Coulomb. The Laguerre polynomials express two sets of solutions, and Bessel functions enter the third one. Also, for these three solutions, we define the exact analytical expressions of the quantum similarities, disequilibria, and entropic moments. In addition, we studied the behavior of quantum dissimilarity, quantum similarity index, and Rényi complexity ratio using every pair of different solutions.Exact solutions of the generalized Dunkl oscillator in the Cartesian systemhttps://zbmath.org/1521.810642023-11-13T18:48:18.785376Z"Dong, Shi-Hai"https://zbmath.org/authors/?q=ai:dong.shihai"Quezada, L. F."https://zbmath.org/authors/?q=ai:quezada.luis-fernando"Chung, W. S."https://zbmath.org/authors/?q=ai:chung.won-sang"Sedaghatnia, P."https://zbmath.org/authors/?q=ai:sedaghatnia.parisa"Hassanabadi, H."https://zbmath.org/authors/?q=ai:hassanabadi.hassanSummary: In this paper, we use the generalized Dunkl derivatives instead of the standard partial derivatives in the Schrödinger equation to obtain an explicit expression of the generalized Dunkl-Schrödinger equation in 3D. It was found that this generalized Dunkl-Schrödinger equation for the 3D harmonic oscillator is exactly solvable in the Cartesian coordinates. From the relevant commutation relations, it is evident that the symmetry possessed by the original Dunkl Harmonic oscillator is \textit{broken} by the generalized Dunkl derivative. Finally, we show that energy levels can be affected by considering a deformation parameter \(\varepsilon\).Ramond-Ramond fields and twisted differential K-theoryhttps://zbmath.org/1521.810672023-11-13T18:48:18.785376Z"Grady, Daniel"https://zbmath.org/authors/?q=ai:grady.daniel"Sati, Hisham"https://zbmath.org/authors/?q=ai:sati.hishamSummary: We provide a systematic approach to describing the Ramond-Ramond (RR) fields as elements in twisted differential K-theory. This builds on a series of constructions by the authors on geometric and computational aspects of twisted differential K-theory, which to a large extent were originally motivated by this problem. In addition to providing a new conceptual framework and a mathematically solid setting, this allows us to uncover interesting and novel effects. Explicitly, we use our recently constructed Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential K-theory to characterize the RR fields and their quantization, which involves interesting interplay between geometric and topological data. We illustrate this with the examples of spheres, tori, and Calabi-Yau threefolds.Duals of Feynman integrals. I: Differential equationshttps://zbmath.org/1521.810872023-11-13T18:48:18.785376Z"Caron-Huot, Simon"https://zbmath.org/authors/?q=ai:caron-huot.simon"Pokraka, Andrzej"https://zbmath.org/authors/?q=ai:pokraka.andrzejSummary: We elucidate the vector space (twisted relative cohomology) that is Poincaré dual to the vector space of Feynman integrals (twisted cohomology) in general spacetime dimension. The pairing between these spaces -- an algebraic invariant called the intersection number -- extracts integral coefficients for a minimal basis, bypassing the generation of integration-by-parts identities. Dual forms turn out to be much simpler than their Feynman counterparts: they are supported on maximal cuts of various sub-topologies (boundaries). Thus, they provide a systematic approach to generalized unitarity, the reconstruction of amplitudes from on-shell data. In this paper, we introduce the idea of dual forms and study their mathematical structures. As an application, we derive compact differential equations satisfied by arbitrary one-loop integrals in non-integer spacetime dimension. A second paper of this series will detail intersection pairings and their use to extract integral coefficients.Wilson loops for 5d and 3d conformal linear quivershttps://zbmath.org/1521.810882023-11-13T18:48:18.785376Z"Fatemiabhari, Ali"https://zbmath.org/authors/?q=ai:fatemiabhari.ali"Nunez, Carlos"https://zbmath.org/authors/?q=ai:nunez.carlosSummary: Within the electrostatic formulation of holographic duals to (balanced) conformal quivers in five and three dimensions, we study the expressions for Wilson loops in antisymmetric representations. We derive general expressions for various quantities participating in the formalism (VEV of Wilson loops, representation, gauge-node) and apply these to examples, connecting some results present in the bibliography. In the case of three dimensional quivers, we present a relation between Wilson loops in an `electric' and in the `magnetic/mirror' descriptions. In a very detailed appendix, we relate the electrostatic and holomorphic description of the holographic duals to these SCFTS.More on Seiberg-Witten theory and monstrous moonshine: a new simple method of calculating the prepotentialhttps://zbmath.org/1521.811072023-11-13T18:48:18.785376Z"Mizoguchi, Shun'ya"https://zbmath.org/authors/?q=ai:mizoguchi.shunya"Oikawa, Takumi"https://zbmath.org/authors/?q=ai:oikawa.takumi"Tashiro, Hitomi"https://zbmath.org/authors/?q=ai:tashiro.hitomi"Yata, Shotaro"https://zbmath.org/authors/?q=ai:yata.shotaroSummary: We continue the study of a relationship between the instanton expansion of the Seiberg-Witten (SW) prepotential of \(D = 4\), \(\mathcal{N} = 2\) \(SU(2)\) SUSY gauge theory and the Monstrous moonshine. As was done in [\textit{S. Mizoguchi}, PTEP, Prog. Theor. Exper. Phys. 2022, No. 12, Article ID 121B01, 9 p. (2022; Zbl 1519.81397)], we expand the inverse of the modular \(j\)-function in \(u^{-1}\) around \(u = \infty\), where \(u\) is the familiar \(u\) parameter for the respective SW curves, and compute the complex gauge coupling \(\tau\) as a function of the scalar vev \(a\) by using the Fourier expansion of \(j(\tau)\) and the relation between \(u\) and \(a\) obtained by the Picard-Fuchs equation. In this way, the instanton expansion of the prepotential is related to the dimensions of representations of the Monster group. We show that, for the cases of \(N_f = 2\) and 3, \(q\) again has an expansion whose coefficients are all integer-coefficient polynomials of the moonshine coefficients if the expansion variables are appropriately chosen. This hints some unknown relation between the Liouville CFT and the vertex operator algebra CFT with different central charges. We also demonstrate that this new method of calculating the SW prepotential developed here is useful by performing some explicit computations.Covariant dynamics on the energy-momentum space: scalar field theoryhttps://zbmath.org/1521.811102023-11-13T18:48:18.785376Z"Ivetić, Boris"https://zbmath.org/authors/?q=ai:ivetic.borisSummary: A scalar field theory is constructed on an energy-momentum background of constant curvature. The generalization of the usual Feynamn rules for the flat geometry follows from the requirement of their covariance. The main result is that the invariant amplitudes are finite at all orders of the perturbation theory, due to the finiteness of the momentum space. Finally, the relation with a field theory in spacetime representation is briefly discussed.Noncommutative geometry of computational models and uniformization for framed quiver varietieshttps://zbmath.org/1521.811172023-11-13T18:48:18.785376Z"Jeffreys, George"https://zbmath.org/authors/?q=ai:jeffreys.george"Lau, Siu-Cheong"https://zbmath.org/authors/?q=ai:lau.siu-cheongSummary: We formulate a mathematical setup for computational neural networks using noncommutative algebras and near-rings, in motivation of quantum automata. We study the moduli space of the corresponding framed quiver representations, and find moduli of Euclidean and non-compact types in light of uniformization.Superconformal algebras and holomorphic field theorieshttps://zbmath.org/1521.811382023-11-13T18:48:18.785376Z"Saberi, Ingmar"https://zbmath.org/authors/?q=ai:saberi.ingmar-a"Williams, Brian R."https://zbmath.org/authors/?q=ai:williams.brian-rSummary: We show that four-dimensional superconformal algebras admit an infinite-dimensional derived enhancement after performing a holomorphic twist. The type of higher symmetry algebras we find is closely related to algebras studied by \textit{G. Faonte} et al. [Adv. Math. 346, 389--466 (2019; Zbl 1409.14027)], \textit{B. Hennion} and \textit{M. Kapranov} [``Gelfand-Fuchs cohomology in algebraic geometry and factorization algebras'', Preprint, \url{arXiv:1811.05032}], and \textit{B. R. Williams} with \textit{O. Gwilliam} [``Higher Kac-Moody algebras and symmetries of holomorphic field theories'', Preprint, \url{arXiv:1810.06534}] in the context of holomorphic QFT. We show that these algebras are related to the two-dimensional chiral algebras extracted from four-dimensional superconformal theories by \textit{C. Beem} et al. [Commun. Math. Phys. 336, No. 3, 1359--1433 (2015; Zbl 1320.81076)]; further deforming by a superconformal element induces the Koszul resolution of a plane in \(\mathbb{C}^2 \cong\mathbb{R}^4\). The central charges at the level of chiral algebras arise from central extensions of the higher symmetry algebras.Sharpening the boundaries between flux landscape and swampland by tadpole chargehttps://zbmath.org/1521.811632023-11-13T18:48:18.785376Z"Ishiguro, Keiya"https://zbmath.org/authors/?q=ai:ishiguro.keiya"Otsuka, Hajime"https://zbmath.org/authors/?q=ai:otsuka.hajimeSummary: We investigate the vacuum structure of four-dimensional effective theory arising from Type IIB flux compactifications on a mirror of the rigid Calabi-Yau threefold, corresponding to a T-dual of the DeWolfe-Giryavets-Kachru-Taylor model in Type IIA flux compactifications. By analyzing the vacuum structure of this interesting corner of string landscape, it turns out that there exist perturbatively unstable de Sitter (dS) vacua in addition to supersymmetric and non-supersymmetric anti-de Sitter vacua. On the other hand, the stable dS vacua appearing in the low-energy effective action violate the tadpole cancellation condition, indicating a strong correlation between the existence of dS vacua and the flux-induced D3-brane charge (tadpole charge). We also find analytically that the tadpole charge constrained by the tadpole cancellation condition emerges in the scalar potential in a nontrivial way. Thus, the tadpole charge would restrict the existence of stable dS vacua, and this fact underlies the statement of the dS conjecture. Furthermore, our analytical and numerical results exhibit that distributions of \(\mathcal{O}(1)\) parameters in expressions of several swampland conjectures peak at specific values.A Laplacian to compute intersection numbers on \(\overline{\mathcal{M}}_{g,n}\) and correlation functions in NCQFThttps://zbmath.org/1521.811852023-11-13T18:48:18.785376Z"Hock, Alexander"https://zbmath.org/authors/?q=ai:hock.alexander"Grosse, Harald"https://zbmath.org/authors/?q=ai:grosse.harald"Wulkenhaar, Raimar"https://zbmath.org/authors/?q=ai:wulkenhaar.raimarSummary: Let \(F_g(t)\) be the generating function of intersection numbers of \(\psi\)-classes on the moduli spaces \(\overline{{{\mathcal{M}}}}_{g,n}\) of stable complex curves of genus \(g\). As by-product of a complete solution of all non-planar correlation functions of the renormalised \(\Phi^3\)-matrical QFT model, we explicitly construct a Laplacian \(\Delta_t\) on a space of formal parameters \(t_i\) which satisfies \(\exp (\sum_{g\ge 2} N^{2-2g}F_g(t))=\exp ((-\Delta_t+F_2(t))/N^2)1\) as formal power series in \(1/N^2\). The result is achieved via Dyson-Schwinger equations from noncommutative quantum field theory combined with residue techniques from topological recursion. The genus-\(g\) correlation functions of the \(\Phi^3\)-matricial QFT model are obtained by repeated application of another differential operator to \(F_g(t)\) and taking for \(t_i\) the renormalised moments of a measure constructed from the covariance of the model.Non-perturbative heterotic duals of M-theory on \(G_2\) orbifoldshttps://zbmath.org/1521.812022023-11-13T18:48:18.785376Z"Acharya, Bobby Samir"https://zbmath.org/authors/?q=ai:acharya.bobby-samir"Kinsella, Alex"https://zbmath.org/authors/?q=ai:kinsella.alex"Morrison, David R."https://zbmath.org/authors/?q=ai:morrison.david-r.1|morrison.david-r.2Summary: By fibering the duality between the \(E_8 \times E_8\) heterotic string on \(T^3\) and M-theory on K3, we study heterotic duals of M-theory compactified on \(G_2\) orbifolds of the form \(T^7/\mathbb{Z}_2^3\). While the heterotic compactification space is straightforward, the description of the gauge bundle is subtle, involving the physics of point-like instantons on orbifold singularities. By comparing the gauge groups of the dual theories, we deduce behavior of a ``half-\(G_2\)'' limit, which is the M-theory analog of the stable degeneration limit of F-theory. The heterotic backgrounds exhibit point-like instantons that are localized on \textit{pairs} of orbifold loci, similar to the ``gauge-locking'' phenomenon seen in Hořava-Witten compactifications. In this way, the geometry of the \(G_2\) orbifold is translated to bundle data in the heterotic background. While the instanton configuration looks surprising from the perspective of the \(E_8 \times E_8\) heterotic string, it may be understood as T-dual Spin(32)/\(\mathbb{Z}_2\) instantons along with winding shifts originating in a dual Type I compactification.D-instantons in type IIA string theory on Calabi-Yau threefoldshttps://zbmath.org/1521.812052023-11-13T18:48:18.785376Z"Alexandrov, Sergei"https://zbmath.org/authors/?q=ai:aleksandrov.sergei-evgenevich|alexandrov.sergei-yu"Sen, Ashoke"https://zbmath.org/authors/?q=au:Sen, Ashoke"Stefański, Bogdan jun."https://zbmath.org/authors/?q=ai:stefanski.bogdan-junSummary: Type IIA string theory compactified on a Calabi-Yau threefold has a hypermultiplet moduli space whose metric is known to receive non-perturbative corrections from Euclidean D2-branes wrapped on 3-cycles. These corrections have been computed earlier by making use of mirror symmetry, S-duality and twistorial description of quaternionic geometries. In this paper we compute the leading corrections in each homology class using a direct world-sheet approach without relying on any duality symmetry or supersymmetry. Our results are in perfect agreement with the earlier predictions.Euclidean D-branes in type IIB string theory on Calabi-Yau threefoldshttps://zbmath.org/1521.812062023-11-13T18:48:18.785376Z"Alexandrov, Sergei"https://zbmath.org/authors/?q=ai:aleksandrov.sergei-evgenevich|alexandrov.sergei-yu"Sen, Ashoke"https://zbmath.org/authors/?q=au:Sen, Ashoke"Stefański, Bogdan jun."https://zbmath.org/authors/?q=ai:stefanski.bogdan-junSummary: We compute the contribution of Euclidean D-branes in type IIB string theory on Calabi-Yau threefolds to the metric on the hypermultiplet moduli space in the large volume, weak coupling limit. Our results are in perfect agreement with the predictions based on S-duality, mirror symmetry and supersymmetry.Hybrid inflation and waterfall field in string theory from D7-braneshttps://zbmath.org/1521.812072023-11-13T18:48:18.785376Z"Antoniadis, Ignatios"https://zbmath.org/authors/?q=ai:antoniadis.ignatios"Lacombe, Osmin"https://zbmath.org/authors/?q=ai:lacombe.osmin"Leontaris, George K."https://zbmath.org/authors/?q=ai:leontaris.george-kSummary: We present an explicit string realisation of a cosmological inflationary scenario we proposed recently within the framework of type IIB flux compactifications in the presence of three magnetised \(D7\)-brane stacks. Inflation takes place around a metastable de Sitter vacuum. The inflaton is identified with the volume modulus and has a potential with a very shallow minimum near the maximum. Inflation ends due to the presence of ``waterfall'' fields that drive the evolution of the Universe from a nearby saddle point towards a global minimum with tuneable vacuum energy describing the present state of our Universe.On M-theory on real toric fibrationshttps://zbmath.org/1521.812102023-11-13T18:48:18.785376Z"Belhaj, A."https://zbmath.org/authors/?q=ai:belhaj.adil"Belmahi, H."https://zbmath.org/authors/?q=ai:belmahi.h"Benali, M."https://zbmath.org/authors/?q=ai:benali.mohammed"Ennadifi, S.-E."https://zbmath.org/authors/?q=ai:ennadifi.salah-eddine"Hassouni, Y."https://zbmath.org/authors/?q=ai:hassouni.yassine"Sekhmani, Y."https://zbmath.org/authors/?q=ai:sekhmani.yassineSummary: Borrowing ideas from elliptic complex geometry, we approach M-theory compactifications on real fibrations. Precisely, we explore real toric equations rather than complex ones exploited in F-theory and related dual models. These geometries have been built by moving real toric manifolds over real bases. Using topological changing behaviors, we unveil certain data associated with gauge sectors relying on affine lie symmetries.D7 moduli stabilization: the tadpole menacehttps://zbmath.org/1521.812112023-11-13T18:48:18.785376Z"Bena, Iosif"https://zbmath.org/authors/?q=ai:bena.iosif"Brodie, Callum"https://zbmath.org/authors/?q=ai:brodie.callum-r"Graña, Mariana"https://zbmath.org/authors/?q=ai:grana.marianaSummary: D7-brane moduli are stabilized by worldvolume fluxes, which contribute to the D3-brane tadpole. We calculate this contribution in the Type IIB limit of F-theory compactifications on Calabi-Yau four-folds with a weak Fano base, and are able to prove a no-go theorem for vast swathes of the landscape of compactifications. When the genus of the curve dual to the D7 worldvolume fluxes is fixed and the number of moduli grows, we find that the D3 charge sourced by the fluxes grows faster than 7/16 of the number of moduli, which supports the Tadpole Conjecture of [\textit{I. Bena} et al., J. High Energy Phys. 2021, No. 11, Paper No. 223, 33 p. (2021; Zbl 1521.81372)]. Our lower bound for the induced D3 charge decreases when the genus of the curves dual to the stabilizing fluxes increase, and does not allow to rule out a sliver of flux configurations dual to high-genus high-degree curves. However, we argue that most of these fluxes have very high curvature, which is likely to be above the string scale except on extremely large (and experimentally ruled out) compactification manifolds.Thraxions: towards full string modelshttps://zbmath.org/1521.812132023-11-13T18:48:18.785376Z"Carta, Federico"https://zbmath.org/authors/?q=ai:carta.federico"Mininno, Alessandro"https://zbmath.org/authors/?q=ai:mininno.alessandro"Righi, Nicole"https://zbmath.org/authors/?q=ai:righi.nicole"Westphal, Alexander"https://zbmath.org/authors/?q=ai:westphal.alexanderSummary: We elucidate various aspects of the physics of thraxions, ultra-light axions arising at Klebanov-Strassler multi-throats in the compactification space of IIB superstring theory. We study the combined stabilization of Kähler moduli and thraxions, showing that under reasonable assumptions, one can solve the combined problem both in a KKLT and a LVS setup. We find that for non-minimal multi-throats, the thraxion mass squared is three-times suppressed by the throat warp factor. However, the minimal case of a double-throat can preserve the six-times suppression as originally found. We also discuss the backreaction of a non-vanishing thraxion vacuum expectation value on the geometry, showing that it induces a breaking of the imaginary self-duality condition for 3-form fluxes. This in turn breaks the Calabi-Yau structure to a complex manifold one. Finally, we extensively search for global models which can accommodate the presence of multiple thraxions within the database of Complete Intersection Calabi-Yau orientifolds. We find that each multi-throat system holds a single thraxion. We further point out difficulties in constructing a full-fledged global model, due to the generic presence of frozen-conifold singularities in a Calabi-Yau orientifold. For this reason, we propose a new database of CICY orientifolds that do not have frozen conifolds but that admit thraxions.Small cosmological constants in string theoryhttps://zbmath.org/1521.812172023-11-13T18:48:18.785376Z"Demirtas, Mehmet"https://zbmath.org/authors/?q=ai:demirtas.mehmet"Kim, Manki"https://zbmath.org/authors/?q=ai:kim.manki"McAllister, Liam"https://zbmath.org/authors/?q=ai:mcallister.liam"Moritz, Jakob"https://zbmath.org/authors/?q=ai:moritz.jakob"Rios-Tascon, Andres"https://zbmath.org/authors/?q=ai:rios-tascon.andresSummary: We construct supersymmetric \(\mathrm{AdS}_4\) vacua of type IIB string theory in compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. We first find explicit orientifolds and quantized fluxes for which the superpotential takes the form proposed by \textit{S. Kachru} et al. [Phys. Rev. D (3) 68, No. 4, Article ID 046005, 10 p. (2003; Zbl 1244.83036)]. Given very mild assumptions on the numerical values of the Pfaffians, these compactifications admit vacua in which all moduli are stabilized at weak string coupling. By computing high-degree Gopakumar-Vafa invariants we give strong evidence that the \(\alpha^\prime\) expansion is likewise well-controlled. We find extremely small cosmological constants, with magnitude \(<10^{-123}\) in Planck units. The compactifications are large, but not exponentially so, and hence these vacua manifest hierarchical scale-separation, with the AdS length exceeding the Kaluza-Klein length by a factor of a googol.Spin(7) orientifolds and 2d \(\mathcal{N} = (0, 1)\) trialityhttps://zbmath.org/1521.812222023-11-13T18:48:18.785376Z"Franco, Sebastián"https://zbmath.org/authors/?q=ai:franco.sebastian"Mininno, Alessandro"https://zbmath.org/authors/?q=ai:mininno.alessandro"Uranga, Ángel M."https://zbmath.org/authors/?q=ai:uranga.angel-m"Yu, Xingyang"https://zbmath.org/authors/?q=ai:yu.xingyangSummary: We present a new, geometric perspective on the recently proposed triality of 2d \(\mathcal{N} = (0, 1)\) gauge theories, based on its engineering in terms of D1-branes probing Spin(7) orientifolds. In this context, triality translates into the fact that multiple gauge theories correspond to the same underlying orientifold. We show how Spin(7) orientifolds based on a particular involution, which we call the universal involution, give rise to precisely the original version of \(\mathcal{N} = (0, 1)\) triality. Interestingly, our work also shows that the space of possibilities is significantly richer. Indeed, general Spin(7) orientifolds extend triality to theories that can be regarded as consisting of coupled \(\mathcal{N} = (0, 2)\) and \((0, 1)\) sectors. The geometric construction of 2d gauge theories in terms of D1-branes at singularities therefore leads to extensions of triality that interpolate between the pure \(\mathcal{N} = (0, 2)\) and \((0, 1)\) cases.Charge completeness and the massless charge lattice in F-theory models of supergravityhttps://zbmath.org/1521.812342023-11-13T18:48:18.785376Z"Morrison, David R."https://zbmath.org/authors/?q=ai:morrison.david-r.1"Taylor, Washington"https://zbmath.org/authors/?q=ai:taylor.washington-ivSummary: We prove that, for every 6D supergravity theory that has an F-theory description, the property of charge completeness for the connected component of the gauge group (meaning that all charges in the corresponding charge lattice are realized by massive or massless states in the theory) is equivalent to a standard assumption made in F-theory for how geometry encodes the global gauge theory by means of the Mordell-Weil group of the elliptic fibration. This result also holds in 4D F-theory constructions for the parts of the gauge group that come from sections and from 7-branes. We find that in many 6D F-theory models the full charge lattice of the theory is generated by massless charged states; this occurs for each gauge factor where the associated anomaly coefficient satisfies a simple positivity condition. We describe many of the cases where this massless charge sufficiency condition holds, as well as exceptions where the positivity condition fails, and analyze the related global structure of the gauge group and associated Mordell-Weil torsion in explicit F-theory models.Taming triangulation dependence of \(T^6/\mathbb{Z}_2\times\mathbb{Z}_2\) resolutionshttps://zbmath.org/1521.813402023-11-13T18:48:18.785376Z"Faraggi, A. E."https://zbmath.org/authors/?q=ai:faraggi.alon-e"Nibbelink, S. Groot"https://zbmath.org/authors/?q=ai:nibbelink.s-groot"Hurtado Heredia, M."https://zbmath.org/authors/?q=ai:hurtado-heredia.mSummary: Resolutions of certain toroidal orbifolds, like \(T^6/\mathbb{Z}_2\times\mathbb{Z}_2\), are far from unique, due to triangulation dependence of their resolved local singularities. This leads to an explosion of the number of topologically distinct smooth geometries associated to a single orbifold. By introducing a parameterisation to keep track of the triangulations used at all resolved singularities simultaneously, (self-)intersection numbers and integrated Chern classes can be determined for any triangulation configuration. Using this method the consistency conditions of line bundle models and the resulting chiral spectra can be worked out for any choice of triangulation. Moreover, by superimposing the Bianchi identities for all triangulation options much simpler though stronger conditions are uncovered. When these are satisfied, flop-transitions between all different triangulations are admissible. Various methods are exemplified by a number of concrete models on resolutions of the \(T^6/\mathbb{Z}_2\times\mathbb{Z}_2\) orbifold.Note on the bundle geometry of field space, variational connections, the dressing field method, \& presymplectic structures of gauge theories over bounded regionshttps://zbmath.org/1521.813412023-11-13T18:48:18.785376Z"François, J."https://zbmath.org/authors/?q=ai:francois.jordan"Parrini, N."https://zbmath.org/authors/?q=ai:parrini.n"Boulanger, N."https://zbmath.org/authors/?q=ai:boulanger.nicolasSummary: In this note, we consider how the bundle geometry of field space interplays with the covariant phase space methods so as to allow to write results of some generality on the presymplectic structure of invariant gauge theories coupled to matter. We obtain in particular the generic form of Noether charges associated with field-independent and field-dependent gauge parameters, as well as their Poisson bracket. We also provide the general field-dependent gauge transformations of the presymplectic potential and 2-form, which clearly highlights the problem posed by boundaries in generic situations. We then conduct a comparative analysis of two strategies recently considered to evade the boundary problem and associate a modified symplectic structure to a gauge theory over a bounded region: namely the use of edge modes on the one hand, and of variational connections on the other. To do so, we first try to give the clearest geometric account of both, showing in particular that edge modes are a special case of a differential geometric tool of gauge symmetry reduction known as the ``dressing field method''. Applications to Yang-Mills theory and General Relativity reproduce or generalise several results of the recent literature.Yukawa textures from singular spectral datahttps://zbmath.org/1521.813472023-11-13T18:48:18.785376Z"Karkheiran, Mohsen"https://zbmath.org/authors/?q=ai:karkheiran.mohsenSummary: The Yukawa textures of effective heterotic models are studied by using singular spectral data. One advantage of this approach is that it is possible to dissect the cohomologies of the bundles into smaller parts and identify the pieces that contain the zero modes, which can potentially have non-zero Yukawa couplings. Another advantage is the manifest relationship between the Yukawa textures in heterotic models and local F-theory models in terms of fields living in bulk or localized inside the 7-branes. We only work with Weierstrass elliptically fibered Calabi-Yau manifolds here. The idea for generalizing this approach to every elliptically fibered Calabi-Yau with rational sections is given at the end of this paper.A hyperbolic analogue of the Atiyah-Hitchin manifoldhttps://zbmath.org/1521.813512023-11-13T18:48:18.785376Z"Sutcliffe, Paul"https://zbmath.org/authors/?q=ai:sutcliffe.paul-m|sutcliffe.paul-jSummary: The Atiyah-Hitchin manifold is the moduli space of parity inversion symmetric charge two SU(2) monopoles in Euclidean space. Here a hyperbolic analogue is presented, by calculating the boundary metric on the moduli space of parity inversion symmetric charge two SU(2) monopoles in hyperbolic space. The calculation of the metric is performed using a twistor description of the moduli space and the result is presented in terms of standard elliptic integrals.Anomalies as obstructions: from dimensional lifts to swamplandhttps://zbmath.org/1521.813562023-11-13T18:48:18.785376Z"Cheng, Peng"https://zbmath.org/authors/?q=ai:cheng.peng.1|cheng.peng|cheng.peng.2"Minasian, Ruben"https://zbmath.org/authors/?q=ai:minasian.ruben"Theisen, Stefan"https://zbmath.org/authors/?q=ai:theisen.stefan-jSummary: We revisit the relation between the anomalies in four and six dimensions and the Chern-Simons couplings one dimension below. While the dimensional reduction of chiral theories is well-understood, the question which three and five-dimensional theories can come from a general circle reduction, and are hence liftable, is more subtle. We argue that existence of an anomaly cancellation mechanism is a necessary condition for liftability. In addition, the anomaly cancellation and the CS couplings in six and five dimensions respectively determine the central charges of string-like BPS objects that cannot be consistently decoupled from gravity, a.k.a. supergravity strings. Following the completeness conjecture and requiring that their worldsheet theory is unitary imposes bounds on the admissible theories. We argue that for the anomaly-free six-dimensional theories it is more advantageous to study the unitarity constraints obtained after reduction to five dimensions. In general these are slightly more stringent and can be cast in a more geometric form, highly reminiscent of the Kodaira positivity condition (KPC). Indeed, for the F-theoretic models which have an underlying Calabi-Yau threefold these can be directly compared. The unitarity constraints (UC) are in general weaker than KPC, and maybe useful in understanding the consistent models without F-theoretic realisation. We catalogue the cases when UC is more restrictive than KPC, hinting at more refined hidden structure in elliptic Calabi-Yau threefolds with certain singularity structure.The tadpole problemhttps://zbmath.org/1521.813722023-11-13T18:48:18.785376Z"Bena, Iosif"https://zbmath.org/authors/?q=ai:bena.iosif"Blåbäck, Johan"https://zbmath.org/authors/?q=ai:blaback.johan"Graña, Mariana"https://zbmath.org/authors/?q=ai:grana.mariana"Lüst, Severin"https://zbmath.org/authors/?q=ai:lust.severinSummary: We examine the mechanism of moduli stabilization by fluxes in the limit of a large number of moduli. We conjecture that one cannot stabilize all complex-structure moduli in F-theory at a generic point in moduli space (away from singularities) by fluxes that satisfy the bound imposed by the tadpole cancellation condition. More precisely, while the tadpole bound in the limit of a large number of complex-structure moduli goes like 1/4 of the number of moduli, we conjecture that the amount of charge induced by fluxes stabilizing all moduli grows faster than this, and is therefore larger than the allowed amount. Our conjecture is supported by two examples: \(K3 \times K3\) compactifications, where by using evolutionary algorithms we find that moduli stabilization needs fluxes whose induced charge is 44\% of the number of moduli, and Type IIB compactifications on \(\mathbb{CP}^3\), where the induced charge of the fluxes needed to stabilize the D7-brane moduli is also 44\% of the number of these moduli. Proving our conjecture would rule out de Sitter vacua obtained via antibrane uplift in long warped throats with a hierarchically small supersymmetry breaking scale, which require a large tadpole.Coulomb branch global symmetry and quiver additionhttps://zbmath.org/1521.813892023-11-13T18:48:18.785376Z"Gledhill, Kirsty"https://zbmath.org/authors/?q=ai:gledhill.kirsty"Hanany, Amihay"https://zbmath.org/authors/?q=ai:hanany.amihaySummary: To date, the best effort made to simply determine the Coulomb branch global symmetry of a theory from a \(3d\) \(\mathcal{N} = 4\) quiver is by applying an algorithm based on its balanced gauge nodes. This often gives the full global symmetry, but there have been many cases seen where it instead gives only a subgroup. This paper presents a method for constructing several families of \(3d\) \(\mathcal{N} = 4\) unitary quivers where the true global symmetry is enhanced from that predicted by the balance algorithm, motivated by the study of Coulomb branch Hasse diagrams. This provides a rich list of examples on which to test improved algorithms for unfailingly identifying the Coulomb branch global symmetry from a quiver.A geometric recipe for twisted superpotentialshttps://zbmath.org/1521.813902023-11-13T18:48:18.785376Z"Hollands, Lotte"https://zbmath.org/authors/?q=ai:hollands.lotte"Rüter, Philipp"https://zbmath.org/authors/?q=ai:ruter.philipp"Szabo, Richard J."https://zbmath.org/authors/?q=ai:szabo.richard-jSummary: We give a pedagogical introduction to spectral networks and abelianization, as well as their relevance to \(\mathcal{N} = 2\) supersymmetric field theories in four dimensions. Motivated by a conjecture of Nekrasov-Rosly-Shatashvili, we detail a geometric recipe for computing the effective twisted superpotential for \(\mathcal{N} = 2\) field theories of class \(\mathcal{S}\) as a generating function of the brane of opers, with respect to the spectral coordinates found from abelianization. We present two new examples, the simplest Argyres-Douglas theory and the pure SU(2) gauge theory, while we conjecture the \(E\)-expansion of the effective twisted superpotential for the \(E_6\) Minahan-Nemeschansky theory.Topological recursion and uncoupled BPS structures. II: Voros symbols and the \(\tau\)-functionhttps://zbmath.org/1521.813942023-11-13T18:48:18.785376Z"Iwaki, Kohei"https://zbmath.org/authors/?q=ai:iwaki.kohei"Kidwai, Omar"https://zbmath.org/authors/?q=ai:kidwai.omarSummary: We continue our study of the correspondence between BPS structures and topological recursion in the uncoupled case, this time from the viewpoint of quantum curves. For spectral curves of hypergeometric type, we show the Borel-resummed Voros symbols of the corresponding quantum curves solve Bridgeland's ``BPS Riemann-Hilbert problem''. In particular, they satisfy the required jump property in agreement with the generalized definition of BPS indices \(\Omega\) in our previous work. Furthermore, we observe the Voros coefficients define a closed one-form on the parameter space, and show that (log of) Bridgeland's \(\tau\)-function encoding the solution is none other than the corresponding potential, up to a constant. When the quantization parameter is set to a special value, this agrees with the Borel sum of the topological recursion partition function \(Z_\mathrm{TR}\), up to a simple factor.
For Part I, see [the authors, Adv. Math. 398, Article ID 108191, 54 p. (2022; Zbl 1486.81157)].Symbol alphabets from tensor diagramshttps://zbmath.org/1521.814012023-11-13T18:48:18.785376Z"Ren, Lecheng"https://zbmath.org/authors/?q=ai:ren.lecheng"Spradlin, Marcus"https://zbmath.org/authors/?q=ai:spradlin.marcus"Volovich, Anastasia"https://zbmath.org/authors/?q=ai:volovich.anastasiaSummary: We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of \(n\)-particle amplitudes in planar \(\mathcal{N} = 4\) Yang-Mills theory and certain polytopes associated to the Grassmannian \(\mathrm{Gr}(4, n)\). We show how to assign a web (a planar tensor diagram) to each facet of these polytopes. Webs with no inner loops are associated to cluster variables (rational symbol letters). For webs with a single inner loop we propose and explicitly evaluate an associated web series that contains information about algebraic symbol letters. In this manner we reproduce the results of previous analyses of \(n \leq 8\), and find that the polytope \(\mathcal{C}^\dagger(4, 9)\) encodes all rational letters, and all square roots of the algebraic letters, of known nine-particle amplitudes.Classical BV formalism for group actionshttps://zbmath.org/1521.814062023-11-13T18:48:18.785376Z"Benini, Marco"https://zbmath.org/authors/?q=ai:benini.marco.1"Safronov, Pavel"https://zbmath.org/authors/?q=ai:safronov.pavel"Schenkel, Alexander"https://zbmath.org/authors/?q=ai:schenkel.alexanderSummary: We study the derived critical locus of a function \(f:[X/G]\to\mathbb{A}_{\mathbb{K}}^1\) on the quotient stack of a smooth affine scheme \(X\) by the action of a smooth affine group scheme \(G\). It is shown that \(\mathrm{dCrit}(f)\simeq[Z/G]\) is a derived quotient stack for a derived affine scheme \(Z\), whose dg-algebra of functions is described explicitly. Our results generalize the classical BV formalism in finite dimensions from Lie algebra to group actions.Gauged 2-form symmetries in 6D SCFTs coupled to gravityhttps://zbmath.org/1521.814762023-11-13T18:48:18.785376Z"Braun, Andreas P."https://zbmath.org/authors/?q=ai:braun.andreas-p"Larfors, Magdalena"https://zbmath.org/authors/?q=ai:larfors.magdalena"Oehlmann, Paul-Konstantin"https://zbmath.org/authors/?q=ai:oehlmann.paul-konstantinSummary: We study six dimensional supergravity theories with superconformal sectors (SCFTs). Instances of such theories can be engineered using type IIB strings, or more generally F-Theory, which translates field theoretic constraints to geometry. Specifically, we study the fate of the discrete 2-form global symmetries of the SCFT sectors. For both \((2, 0)\) and \((1, 0)\) theories we show that whenever the charge lattice of the SCFT sectors is non-primitively embedded into the charge lattice of the supergravity theory, there is a subgroup of these 2-form symmetries that remains unbroken by BPS strings. By the absence of global symmetries in quantum gravity, this subgroup much be gauged. Using the embedding of the charge lattices also allows us to determine how the gauged 2-form symmetry embeds into the 2-form global symmetries of the SCFT sectors, and we present several concrete examples, as well as some general observations. As an alternative derivation, we recover our results for a large class of models from a dual perspective upon reduction to five dimensions.The characteristic initial value problem for the conformally invariant wave equation on a Schwarzschild backgroundhttps://zbmath.org/1521.830212023-11-13T18:48:18.785376Z"Hennig, Jörg"https://zbmath.org/authors/?q=ai:hennig.jorg-dieterSummary: We resume former discussions of the conformally invariant wave equation on a Schwarzschild background, with a particular focus on the behaviour of solutions near the `cylinder', i.e. Friedrich's representation of spacelike infinity. This analysis can be considered a toy model for the behaviour of the full Einstein equations and the resulting logarithmic singularities that appear to be characteristic for massive spacetimes. The investigation of the \textit{Cauchy} problem for the conformally invariant wave equation \textit{J. Frauendiener} and \textit{J. Hennig} [Classical Quantum Gravity 35, No. 6, Article ID 065015, 19 p. (2018; Zbl 1386.83080)] showed that solutions generically develop logarithmic singularities at infinitely many expansion orders at the cylinder, but an arbitrary finite number of these singularities can be removed by appropriately restricting the initial data prescribed at \(t = 0\). From a physical point of view, any data at \(t = 0\) are determined from the earlier history of the system and hence not exactly `free data'. Therefore, it is appropriate to ask what happens if we `go further back in time' and prescribe initial data as early as possible, namely at a portion of past null infinity, and on a second past null hypersurface to complete the initial value problem. Will regular data at past null infinity automatically lead to a regular evolution up to future null infinity? Or does past regularity restrict the solutions too much, and regularity at both null infinities is mutually exclusive? Or do we still have suitable degrees of freedom for the data that can be chosen to influence regularity of the solutions to any desired degree? In order to answer these questions, we study the corresponding \textit{characteristic} initial value problem. In particular, we investigate in detail the appearance of singularities at expansion orders \(n = 0, \dots, 4\) for angular modes \(\ell = 0, \dots, 4\).Extended phase space thermodynamics of black holes: a study in Einstein's gravity and beyondhttps://zbmath.org/1521.830902023-11-13T18:48:18.785376Z"Bhattacharya, Krishnakanta"https://zbmath.org/authors/?q=ai:bhattacharya.krishnakantaSummary: In the extended phase space approach, one can define thermodynamic pressure and volume that gives rise to the van der Waals type phase transition for black holes. For Einstein's GR, the expressions of these quantities are unanimously accepted. Of late, the van der Waals phase transition in black holes has been found in modified theories of gravity as well, such as the \(f(R)\) gravity and the scalar-tensor gravity. However, in the case of these modified theories of gravity, the expression of pressure (and, hence, volume) is not uniquely determined. In addition, for these modified theories, the extended phase space thermodynamics has not been studied extensively, especially in a covariant way. Since both the scalar-tensor and the \(f(R)\) gravity can be discussed in the two conformally connected frames (the Jordan and the Einstein frame respectively), the arbitrariness in the expression of pressure, will act upon the equivalence of the thermodynamic parameters in the two frames. We highlight these issues in the paper. Before that, in Einstein's gravity (GR), we obtain a general expression of the equilibrium state version of first law and the Smarr-like formula from the Einstein's equation for a general static and spherically symmetric (SSS) metric. Unlike the existing formalisms in literature which defines thermodynamic potential in order to express the first law, here we directly obtain the first law as well as the Smarr-like formula in GR in terms of the parameters present in the metric (such as mass, charge \textit{etc.}). This study also shows how the extended phase space is formulated (by considering the cosmological constant as variable) and, also shows why the cosmological constant plays the role of thermodynamic pressure in GR in extended phase space. Moreover, obtaining the Smarr formula from the Einstein's equation for the SSS metric suggests that this dynamical equation encodes more information on BH thermodynamics than what has been anticipated before.Scale-invariance at the core of quantum black holeshttps://zbmath.org/1521.830932023-11-13T18:48:18.785376Z"Borissova, Johanna N."https://zbmath.org/authors/?q=ai:borissova.johanna-n"Held, Aaron"https://zbmath.org/authors/?q=ai:held.aaron"Afshordi, Niayesh"https://zbmath.org/authors/?q=ai:afshordi.niayeshSummary: We study spherically-symmetric solutions to a modified Einstein-Hilbert action with renormalization group (RG) scale-dependent couplings, inspired by Weinberg's Asymptotic Safety scenario for quantum gravity. The RG scale is identified with the Tolman temperature for an isolated gravitational system in thermal equilibrium with Hawking radiation. As a result, the point of infinite local temperature is shifted from the classical black-hole horizon to the origin and coincides with a timelike curvature singularity. Close to the origin, the spacetime is determined by the scale-dependence of the cosmological constant in the vicinity of the Reuter fixed point: the free components of the metric can be derived analytically and are characterized by a radial power law with exponent \(\alpha = \sqrt{3} - 1\). Away from the fixed point, solutions for different masses are studied numerically and smoothly interpolate between the Schwarzschild exterior and the scale-invariant interior. Whereas the exterior of objects with astrophysical mass is described well by vacuum general relativity, deviations become significant at a Planck distance away from the classical horizon and could lead to observational signatures. We further highlight potential caveats in this intriguing result with regard to our choice of scale-identification and identify future avenues to better understand quantum black holes in relation to the key feature of scale-invariance.The special role of toroidal black holes in holographyhttps://zbmath.org/1521.831422023-11-13T18:48:18.785376Z"McInnes, Brett"https://zbmath.org/authors/?q=ai:mcinnes.brettSummary: In the standard holographic ``dictionary'', the deep infrared of the strongly coupled boundary field theory is studied by examining the bulk region near to the event horizon of a simple AdS-Reissner-Nordström black hole, near to extremality. Recently Horowitz et al. have argued that this is not correct, \textit{except} in the case of small toroidal black holes, which are therefore revealed to be particularly interesting and important. On the other hand, the Weak Gravity Conjecture postulates that black holes (including toroidal black holes) which are extremely near to extremality spontaneously emit black holes of the same kind. We show that, in the toroidal case, these ``emitted'' black holes are always small in the sense of Horowitz et al. As an application, we discuss the Grinberg-Maldacena analysis of the way one-point functions, evaluated outside an AdS-Reissner-Nordström black hole, depend on the proper time of fall from the event horizon to the Cauchy horizon. We find that, for emitted toroidal black holes, this dependence effectively drops out.Attractors with large complex structure for one-parameter families of Calabi-Yau manifoldshttps://zbmath.org/1521.831762023-11-13T18:48:18.785376Z"Candelas, Philip"https://zbmath.org/authors/?q=ai:candelas.philip"Kuusela, Pyry"https://zbmath.org/authors/?q=ai:kuusela.pyry"McGovern, Joseph"https://zbmath.org/authors/?q=ai:mcgovern.josephSummary: The attractor equations for an arbitrary one-parameter family of Calabi-Yau manifolds are studied in the large complex structure region. These equations are solved iteratively, generating what we term an \(N\)-expansion, which is a power series in the Gromov-Witten invariants of the manifold. The coefficients of this series are associated with integer partitions. In important cases we are able to find closed-form expressions for the general term of this expansion. To our knowledge, these are the first generic solutions to attractor equations that incorporate instanton contributions. In particular, we find a simple closed-form formula for the entropy associated to rank two attractor points, including those recently discovered. The applications of our solutions are briefly discussed. Most importantly, we are able to give an expression for the Wald entropy of black holes that includes all genus 0 instanton corrections.Modular curves and the refined distance conjecturehttps://zbmath.org/1521.831852023-11-13T18:48:18.785376Z"Kläwer, Daniel"https://zbmath.org/authors/?q=ai:klawer.danielSummary: We test the refined distance conjecture in the vector multiplet moduli space of 4D \(\mathcal{N} = 2\) compactifications of the type IIA string that admit a dual heterotic description. In the weakly coupled regime of the heterotic string, the moduli space geometry is governed by the perturbative heterotic dualities, which allows for exact computations. This is reflected in the type IIA frame through the existence of a K3 fibration. We identify the degree \(d = 2N\) of the K3 fiber as a parameter that could potentially lead to large distances, which is substantiated by studying several explicit models. The moduli space geometry degenerates into the modular curve for the congruence subgroup \(\Gamma_0(N)^+\). In order to probe the large \(N\) regime, we initiate the study of Calabi-Yau threefolds fibered by general degree \(d > 8\) K3 surfaces by suggesting a construction as complete intersections in Grassmann bundles.Two new classes of Hermitian self-orthogonal non-GRS MDS codes and their applicationshttps://zbmath.org/1521.941222023-11-13T18:48:18.785376Z"Luo, Gaojun"https://zbmath.org/authors/?q=ai:luo.gaojun"Cao, Xiwang"https://zbmath.org/authors/?q=ai:cao.xiwang"Ezerman, Martianus Frederic"https://zbmath.org/authors/?q=ai:ezerman.martianus-frederic"Ling, San"https://zbmath.org/authors/?q=ai:ling.sanSummary: Hermitian self-orthogonal codes form an essential ingredient in the construction of quantum stabilizer codes. In this paper, we present two new classes of Hermitian self-orthogonal MDS codes from twisted generalized Reed-Solomon codes. The constructed codes are monomially inequivalent to generalized Reed-Solomon codes. Two new classes of Hermitian LCD MDS codes can then be derived. Finally, based on the constructed Hermitian self-orthogonal MDS codes, we present two classes of quantum MDS codes with new parameters.