Recent zbMATH articles in MSC 14Jhttps://zbmath.org/atom/cc/14J2023-09-22T14:21:46.120933ZWerkzeugRamanujan-type systems of nonlinear ODEs for \(\Gamma_0 (2)\) and \(\Gamma_0 (3)\)https://zbmath.org/1517.110322023-09-22T14:21:46.120933Z"Nikdelan, Younes"https://zbmath.org/authors/?q=ai:nikdelan.younesSummary: This paper aims to introduce two systems of nonlinear ordinary differential equations whose solution components generate the graded algebra of quasi-modular forms on Hecke congruence subgroups \(\Gamma_0 (2)\) and \(\Gamma_0 (3)\). Using these systems, we provide the generated graded algebras with an \(\mathfrak{sl}_2 (\mathbb{C})\)-module structure. As applications, we introduce Ramanujan-type tau functions for \(\Gamma_0 (2)\) and \(\Gamma_0 (3)\), and obtain some interesting and non-trivial recurrence and congruence relations.Hilbert basis resolutions for three-dimensional canonical cyclic quotient singularitieshttps://zbmath.org/1517.140062023-09-22T14:21:46.120933Z"Sato, Kohei"https://zbmath.org/authors/?q=ai:sato.kohei"Sato, Yusuke"https://zbmath.org/authors/?q=ai:sato.yusukeSummary: \textit{M.-N. Ishida} and \textit{N. Iwashita} classified three-dimensional canonical cyclic quotient singularities [Adv. Stud. Pure Math. 8, 135--151 (1987; Zbl 0627.14002)]. By using the classification, we shall discuss Hilbert basis resolutions via \(\operatorname{Hilb}^G(\mathbb{C}^3)\), Fujiki-Oka resolutions and iterated Fujiki-Oka resolutions. In particular, we shall prove that there exists a Hilbert basis resolution for any three-dimensional canonical cyclic quotient singularity.
For the entire collection see [Zbl 1516.14004].Kummer quartic double solidshttps://zbmath.org/1517.140092023-09-22T14:21:46.120933Z"Cheltsov, Ivan"https://zbmath.org/authors/?q=ai:cheltsov.ivanKummer quartic surfaces are those irreducible normal quartic surfaces in \(\mathbb P^3\) with 16 ordinary double points (the maximal number of singular points). Such a quartic surface \(\mathcal S\) is the Kummer variety of the Jacobian surface of a smooth genus \(2\) curve \(\mathcal C\). There is a rich correspondence between the surface \(\mathcal S\) and the hyperelliptic curve \(\mathcal C\). In particular, the group of automorphisms of \(\mathbb P^3\) leaving \(\mathcal S\) invariant, \(\mathrm{Aut} (\mathbb P^3, \mathcal S)\), can be expressed in terms of the quotient of the automorphism group of \(\mathcal C\) by the hyperelliptic involution.
The main focus of the article are Kummer quartic double solids. These are double covers \(\pi \colon X\to \mathbb P^3\) branched along a Kummer quartic surface \(\mathcal S\), and they are del Pezzo threefolds of degree \(2\). The main result of the paper states optimal conditions under which \(X\) is \(G\)-birationally super-rigid, where \(G\) is a subgroup of \(\mathrm{Aut} (X)\). The paper provides methods to check geometrically whether these conditions hold, and constructs \(G\)-Sarkisov links violating birational super-rigid when they are not.
The article presents a large number of examples and explores their birational geometry in detail.
Reviewer: Anne-Sophie Kaloghiros (London)Wall-crossing for iterated Hilbert schemes (or `Hilb of Hilb')https://zbmath.org/1517.140132023-09-22T14:21:46.120933Z"Wormleighton, Ben"https://zbmath.org/authors/?q=ai:wormleighton.benSummary: We study wall-crossing phenomena in the McKay correspondence. Craw-Ishii show that every projective crepant resolution of a Gorenstein abelian quotient singularity arises as a moduli space of \(\theta \)-stable representations of the McKay quiver. The stability condition \(\theta\) moves in a vector space with a chamber decomposition in which (some) wall-crossings capture flops between different crepant resolutions. We investigate where chambers for certain resolutions with Hilbert scheme-like moduli interpretations -- iterated Hilbert schemes, or `Hilb of Hilb' - sit relative to the principal chamber defining the usual \(G\)-Hilbert scheme. We survey relevant aspects of wall-crossing, pose our main conjecture, prove it for some examples and special cases, and discuss connections to other parts of the McKay correspondence.
For the entire collection see [Zbl 1516.14004].Relating derived equivalences for simplices of higher-dimensional flopshttps://zbmath.org/1517.140162023-09-22T14:21:46.120933Z"Donovan, Will"https://zbmath.org/authors/?q=ai:donovan.willSummary: I study a sequence of singularities in dimension 4 and above, each given by a cone of rank 1 tensors of a certain signature, which have crepant resolutions whose exceptional loci are isomorphic to cartesian powers of the projective line. In each dimension \(n\), these resolutions naturally correspond to vertices of an \((n - 2)\)-simplex, and flops between them correspond to edges of the simplex. I show that each face of the simplex may then be associated to a certain relation between flop functors.
For the entire collection see [Zbl 1516.14004].Families of stable 3-folds in positive characteristichttps://zbmath.org/1517.140232023-09-22T14:21:46.120933Z"Kollár, János"https://zbmath.org/authors/?q=ai:kollar.janosOver fields of characteristic 0, the moduli space of stable varieties has been recently shown to be a proper DM stack by several authors (see the book [\textit{J. Kollár}, Families of varieties of general type. With the collaboration of Klaus Altmann and Sándor J. Kovács. Cambridge: Cambridge University Press (2023; Zbl 07658187)] for a comprehensive treatment of the theory).
The properness of the moduli space in characteristic 0 crucially relies on the existence of stable limits for families of stable varieties, which depends on the Minimal Model Program and a local Kawamata-Viehweg vanishing theorem for slc singularities first established by \textit{V. Alexeev} [Pure Appl. Math. Q. 4, No. 3, 767--783 (2008; Zbl 1158.14020)].
Despite the recent advances in characteristic 0, the theory is at its early stages in characteristic \(p>0\). In this article, Kollár shows that slc (in particular, \(S_2\)) stable 3-folds do not form a proper moduli space in positive characteristic (Theorem 1).
More precisely, the author constructs in every characteristic \(p>0\) a family of dlt 3-dimensional pairs with big and nef canonical divisor for which infinitely many plurigenera jump on the central fibre. This shows that the central fibre of the canonical model of the family is not \(S_2\), and thus it does not coincide with the canonical model of the central fibre (Example 4). The construction relies on an example showing the failure of invariance of plurigenera for families of surface pairs of Kodaira dimension 1 (Example 8). From the singularity point of view, this shows that the local Kawamata-Viehweg mentioned above fails for slc 4-dimensional singularities in every positive characteristic and that there exists a log canonical centre which is not weakly normal (Corollary 5. Additional examples in higher dimensions are also considered.
Reviewer: Fabio Bernasconi (Lausanne)Discrete parametric surfaceshttps://zbmath.org/1517.140242023-09-22T14:21:46.120933Z"Wallner, Johannes"https://zbmath.org/authors/?q=ai:wallner.johannes|wallner.johannes-peterSummary: Discrete parametric surfaces are discrete analogues of smooth parametric surfaces. They are, however, not simply discrete approximations of their smooth counterparts, but are the subject of a separate discrete theory. As it turns out, a systematic theory of parametric surfaces can be based on integrable systems, and the discrete case can be interpreted as a ``master'' case which contains smooth surfaces as a limit.
This section on discrete parametric surfaces is organized as follows: We first introduce notation. Two particular kinds of discrete surfaces are discussed next: circular nets in \S2, and K-nets in \S3 are examples of a 3-system and a 2-system, respectively. In the case of K-nets, we also discuss the relation to the sine-Gordon equation. We then show applications within mathematics in \S4, cf. [\textit{A. I. Bobenko} et al., Ann. Math. (2) 164, No. 1, 231--264 (2006; Zbl 1122.53003)], and the connection with freeform architecture in \S5, cf. [\textit{H. Pottmann} and \textit{J. Wallner}, in: Mathematics and society. Zürich: European Mathematical Society (EMS). 131--151 (2016; Zbl 1354.00063)]. The main source for this chapter is the monograph [\textit{A. I. Bobenko} and \textit{Y. B. Suris}, Discrete differential geometry. Integrable structure. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1158.53001)].
For the entire collection see [Zbl 1455.53045].On the cohomology groups of real Lagrangians in Calabi-Yau threefoldshttps://zbmath.org/1517.140252023-09-22T14:21:46.120933Z"Argüz, Hülya"https://zbmath.org/authors/?q=ai:arguz.hulya"Prince, Thomas"https://zbmath.org/authors/?q=ai:prince.thomas-mThis article is concerned with the study of the mod 2 and integral cohomology groups of a real Lagrangian \(L\), obtained as the fixed locus of an anti-symplectic involution in the mirror to \(X\), a Calabi-Yau threefold. First, the authors prove some topological properties of \(L\). Indeed they prove that each of its connected components is orientable and it is the disjoint union of a 3-sphere and a rational homology sphere.
Then the authors use the a Čech-to-derived spectral sequence to deduce a correspondence between the mod 2 Betti numbers of \(L\) and the set of integral points in the reflexive polytope \(P\) that defines \(X\). Doing so, the authors identify the mod 2 Betti numbers of \(L\) with certain Hodge numbers of \(X\).
In the last section they give a more topological, approach to the problem finding that it applies in much greater generality. They consider the case where \(X\) and \(X'\) form a Batrev-Borisov mirror pair. As in the quintic case studied in this article, the real Lagrangian \(L\subset X'\) is the disjoint union of a 3-sphere and a nontrivial component \(L'\). Then, they construct a Heegaard splitting of \(L' R\) and explain with an algorithm how to use this splitting to compute \(\pi(L')\). As this determines its first cohomology group with integer coefficients this verifies parts of the Čech-to-derived calculations done in the previous sections, so they conjecture a more general result, which is the following.
Let \(X\) be Calabi-Yau 3-fold obtained as a (crepant resolution of a) complete intersection in a toric Fano variety. Let \(f : X \to B\) be a Lagrangian torus fibration and let \(L\) be the real Lagrangian in \(X\) obtained as fixed-point locus of the anti-symplectic involution constructed in [\textit{R. Castaño-Bernard} et al., Adv. Math. 225, No. 3, 1341--1386 (2010; Zbl 1203.53085)] and given on smooth fibers of \(f\) by \(x\mapsto -x\). Then, the cohomology groups \(H^j (L,Z)\) are 2-primary for \(0 < j < 3\).
Reviewer: Lucas Li Bassi (Poitiers)Deformations of rational curves on primitive symplectic varieties and applicationshttps://zbmath.org/1517.140262023-09-22T14:21:46.120933Z"Lehn, Christian"https://zbmath.org/authors/?q=ai:lehn.christian"Mongardi, Giovanni"https://zbmath.org/authors/?q=ai:mongardi.giovanni"Pacienza, Gianluca"https://zbmath.org/authors/?q=ai:pacienza.gianlucaPrimitive or irreducible symplectic varieties are the ``singular'' analogs of holomorphic symplectic varieties. Examples of symplectic varieties with singularities are moduli spaces \(M_v(S, \sigma)\) of \(\sigma\)-semistable objects of Mukai vector \(v\) on a projective \(K3\) surface \(S\), or fibers \(K_v(S, \sigma)\) of the Albanese map on \(M_v(S, \sigma)\) when \(S\) is an abelian surface, where \(\sigma\) is a \(v\)-generic stability condition, see [\textit{A. Bayer} and \textit{E. Macrì}, J. Am. Math. Soc. 27, No. 3, 707--752 (2014; Zbl 1314.14020); \textit{H. Minamide} et al., Int. Math. Res. Not. 2014, No. 19, 5264--5327 (2014; Zbl 1327.14087)] for references. In particular, Proposition 2.15 characterizes when the moduli spaces are primitive symplectic varieties, c.f. [\textit{A. Perego} and \textit{A. Rapagnetta}, Algebr. Geom. 10, No. 3, 348--393 (2023; Zbl 07686344)].
Together with F.~Charles, the second and third authors studied the deformation of rational curves on holomorphic symplectic varieties. Let \(X\) be a holomorphic symplectic variety of dimension \(2n\), let \(f \colon C\to X\) be a stable map of a genus zero curve \(C\), and let \(\mathscr{X}\to S\) be a smooth projective family of holomorphic symplectic varieties such that \(\mathscr{X}_0=X\). They proved that
\begin{itemize}
\item[1.] Any irreducible component \(M\) of the Kontsevich moduli stack \(\overline{M}_0(X, f_*[C])\) has dimension at least \(2n-2\).
\item[2.] If \(\dim M=2n-2\), then the stable map \(f\colon C\to X\), deforms along the Hodge locus \(B_\alpha\subset S\) associated to the Hodge class \(\alpha:=f_*[C]\in H^{2n-2}(X)\).
\end{itemize}
A divisor \(D\subset X\) is called uniruled if there exists a family \(p: \mathscr{C}\to T\) of stable curves of genus \(0\) such that \(D\) is the image of the evaluation map \(\operatorname{ev}: \mathscr{C}\to X\). We say a genus zero stable curve \(f\colon C\to X\) is the ruling of \(p\) if \(C=p^{-1}(0)\). When \(f\) is ruling, they showed that \(\dim M=2n-2\), which implies that a uniruled divisor in a holomorphic symplectic variety can be deformed along the Hodge locus associated to the cycle class \(f_*[C]\). Moreover, under the duality \(H^2(X, \mathbb{Z})\cong H_2(X, \mathbb{Z})\) with respect to the Beauville-Bogomolov quadratic form, the cohomology class of the uniruled divisor \([D]\) is proportional to the dual of the curve class \(f_*[C]\).
In the paper under review, Theorem 1.1 extends the above conclusions to primitive symplectic varieties under locally trivial deformations.
In theorem 2.12 and Corollary 2.13, the authors prove that the moduli space \(M_v(S, \sigma)\) (resp. \(K_v(S, \sigma)\)), with some numerical assumption, is locally trivial deformation equivalent to the moduli space \(M_v(S', H)\) (resp. \(K_v(S',H)\)) of stable sheaves of Mukai vector \(v:=(0, mH, 0)\) on a \(K3\) (resp. abelian) surface \(S'\) with a primitive ample divisor \(H\) and \(m\geq 1\). In Proposition 4.4 and 4.5, using the construction of [\textit{A. Perego} and \textit{A. Rapagnetta}, Algebr. Geom. 10, No. 3, 348--393 (2023; Zbl 07686344)], it is shown that \(M_v(S', H)\) and \(K_v(S', H)\) contains infinitely many non-propotional uniruled ample divisors of different positive squares.
For any moduli space \(\mathfrak{M}\) of polarized primitive symplectic varieties that deformation equivalent to \(M_v(S, \sigma)\), the infinitely many uniruled ample divisors \(\{D\}\) on \(M_v(S', H)\) corresponds to infinitely many connected components \(\mathfrak{M}_h\) of \(\mathfrak{M}\) whose polarization \(h\) corresponds to each unirulde divisor \(D\). As a consequence of Theorem 1.1, Theorem 1.3 asserts that any primitive symplectic variety in \(\mathfrak{M}_h\) contains a uniruled divisor that is proportional to the polarization \(h\).
Reviewer: Renjie Lyu (Beijing)Fano fourfolds having a prime divisor of Picard number 1https://zbmath.org/1517.140272023-09-22T14:21:46.120933Z"Secci, Saverio Andrea"https://zbmath.org/authors/?q=ai:secci.saverio-andreaA Fano variety over \(\mathbb{C}\) is a projective manifold whose anti-canonical divisor is ample. Due to a series of works by Iskovskih and Mori-Mukai, there is a complete classification result for Fano varieties up to dimension 3. However, a full classification of Fano varieties is currently out of reach if the dimension is greater than 3. In the paper under review, the author proves a classification result for smooth complex Fano fourfolds of Picard number 3 having a prime divisor of Picard number 1. Based on a classification result in arbitrary dimension by \textit{C. Casagrande} and \textit{S. Druel} [Int. Math. Res. Not. 2015, No. 21, 10756--10800 (2015; Zbl 1342.14088)], the author shows that such fourfolds form 28 families (see Theorem 1.1). The author computes their numerical invariants, determine the base locus of the anticanonical system and study their deformations to give an upper bound to the dimension of the base of the Kuranishi family of a general member (see Tables 2 and 3).
Reviewer: Guolei Zhong (Daejeon)A note on the existence of certain rank 2 stable bundles on very general hypersurfaces of degree at least 5 in \(\mathbb{P}^3\)https://zbmath.org/1517.140282023-09-22T14:21:46.120933Z"Bhattacharya, Debojyoti"https://zbmath.org/authors/?q=ai:bhattacharya.debojyoti.1This short note studies the existence of (slope) stable rank 2 vector bundles \(E\) such that \(h^0(E)\geq k+1\), in other words, the non-emptiness of certain Brill-Noether loci \(\mathcal{B}_2^k\) in the moduli space \(M_H(2;c_1,c_2)\) of \(H\)-semistable sheaves on a very general hypersurface \(X\subset\mathbb{P}^3\) of degree \(t\geq5\).
The author constructs the required vector bundles in two different ways: the Mukai exact sequence and the Serre correspondence.
In the first case, the author chooses a smooth curve \(C\) on \(X\) with degree \(t-2+n\), i.e. \(C\in |(t-2+n)H|\) with \(H=\mathcal{O}_{\mathbb{P}^3}(1)\otimes\mathcal{O}_X\). Denote by \(g_C=\frac{t(t-2+n)(2t-6+n)}2+1\) the genus of \(C\). Then the author shows that for \(d\in\{g_c-3,g_c-2,g_c-1,g_c,g_c+1\}\), it is possible to find \(L\in \mathrm{Pic}^d(C)\) such that \(L\) is base point free, \(h^0(L)=2\) and \(W_d^1(C):=\{\widetilde{L}\in \mathrm{Pic}^d(C)|h^0(\widetilde{L})\geq k+1\}\) is smooth of expected dimension at \(L\). With this chosen \((C,L)\), one has the Mukai exact sequence
\[
0\rightarrow \mathcal{K}\rightarrow H^0(C,L)\otimes \mathcal{O}_X\xrightarrow{ev} L\rightarrow 0,
\]
where the dual \(\mathcal{K}^{-1}\) is a rank 2 slop stable bundle with \(c_1(\mathcal{K}^{-1})=(t-2+n)H\) and \(c_2(\mathcal{K}^{-1})=\deg~L\). Therefore the Brill-Noether loci \(\mathcal{B}_2^1-\mathcal{B}_2^2\subset M_H(2,(t-2+n)H,c_2)\) with \(g_c-3\leq c_2\leq g_c+1 \) are not empty.
The second case seems even simpler: choose \(P\in \mathrm{Hilb}^{c_2}(X)\) with \(c_2>\binom{2t-3+n}{3}-\binom{t-3+n}{3}\), to be a general point and to satisfy that the corresponding zero dimensional subscheme is a locally complete intersection. By Serre correspondence the middle term \(E\) in
\[
0\rightarrow \mathcal{O}_X\rightarrow E\rightarrow \mathcal{I}_P(t-2-n)\rightarrow 0
\]
is a rank 2 slope stable bundle. Hence for \(n>t-6\), \(E\in \mathcal{B}_2^0\subset M_H(2;(t-2+n)H,c_2)\).
Most results of this paper are direct generalizations of existing results due to others. Some proofs are even omitted as they are almost the same as in the work cited.
Reviewer: Yao Yuan (Beijing)Ulrich bundles on cubic fourfoldshttps://zbmath.org/1517.140292023-09-22T14:21:46.120933Z"Faenzi, Daniele"https://zbmath.org/authors/?q=ai:faenzi.daniele"Kim, Yeongrak"https://zbmath.org/authors/?q=ai:kim.yeongrakThe authors tackle the Ulrich existence problem and the Ulrich complexity problem by considering a smooth cubic fourfold \(X\) i.e. degree \(d=3\) and dimension \(n=4\) and prove that \(X\) admits an Ulrich bundle \(\mathfrak{U}\) of rank 6. The authors show the existence of rank 6 Ulrich bundles on smooth cubic fourfolds by first constructing a simple sheaf \(\mathscr{E}\) as an elementary modification of an arithmetically Cohen-Macaulay (aCM) bundle of rank 6 to get the right cubic polynomial and since \(\mathscr{E}\) is not an Ulrich bundle but a general deformation of \(\mathscr{E}(1)\) is Ulrich.
Reviewer: Damian Maingi (Nairobi)Birational geometry and the canonical ring of a family of determinantal 3-foldshttps://zbmath.org/1517.140302023-09-22T14:21:46.120933Z"Lazić, Vladimir"https://zbmath.org/authors/?q=ai:lazic.vladimir"Schreyer, Frank-Olaf"https://zbmath.org/authors/?q=ai:schreyer.frank-olafThis article studies in detail the geometry of a class of determinantal varieties. The varieties considered are defined as the locus in a product of projective spaces where a general homomorphism \(\varphi \colon \mathcal F\to \mathcal G\) between vector bundles of ranks \(f\leq g\) fails to have maximal rank.
The authors study examples where \(\mathbb P = \mathbb P^2\times \mathbb P^3\), and the vector bundles \(\mathcal F\) and \(\mathcal G\) have ranks \(2\) and \(3\) respectively. In this case, \(X\) is a smooth threefold. The article focuses on the class of examples \(X_b\) obtained by taking \(\mathcal F= \mathcal O^{\oplus 2}\), and
\[
\mathcal G_b= \mathcal O_{\mathbb P}(1,b) \oplus \ker \big(H^0(\mathbb P, \mathcal O_{\mathbb P}(1,0)))\otimes \mathcal O_{\mathbb P}(1,1) \to O_{\mathbb P}(2,1)\big)
\]
and \(\varphi\) the evaluation morphism in suitable coordinates on \(\mathbb P^2\).
After studying cohomological properties of \(X_b\), the authors describe its birational geometry. They construct and study in detail a birational map from \(X_b\) to a hypersurface \(Y_b\subset \mathbb P(1^4, b+1)\) of degree \(2b+2\). It follows that \(X_b\) has Kodaira dimension \(-\infty\) when \(b= 1,2\), \(0\) when \(b=3\) and \(3\) when \(b\geq 4\). When \(b\geq 4\), \(X_b\) is of general type, \(Y_b\) is its canonical model and the geometric construction exhibits a minimal model.
Reviewer: Anne-Sophie Kaloghiros (London)On centres and direct sum decompositions of higher degree formshttps://zbmath.org/1517.150172023-09-22T14:21:46.120933Z"Huang, Hua-Lin"https://zbmath.org/authors/?q=ai:huang.hua-lin"Lu, Huajun"https://zbmath.org/authors/?q=ai:lu.huajun"Ye, Yu"https://zbmath.org/authors/?q=ai:ye.yu"Zhang, Chi"https://zbmath.org/authors/?q=ai:zhang.chi.4The authors show that almost all direct sum decompositions of higher degree forms have trivial centre, i.e., isomorphic to the ground field. This means that they are a priori absolutely indecomposable. They also prove that the centre of the algebra of a higher-degree form is semisimple iff the form is not a limit of direct sums forms, as discussed in [\textit{W. Buczyńska} et al., Mich. Math. J. 64, No. 4, 675--719 (2015; Zbl 1339.13012)]. Furthermore, for forms with semisimple centre the authors develop an elementary criterion for the direct sum decomposability, which is equivalent to computing the rank of a finite set of vectors.
In Theorem 3.2, it is shown that a generic higher degree form is central; with the help of Theorems 3.6 and 3.7, a central form may be expressed as an indecomposable non-LDS (here LDS means limit of direct sums). In Theorem 3.11, an algorithm is discussed, purely in terms of linear algebra, for direct sum decompositions of any higher degree form. Such an algorithm can be obtained by slightly extending the algorithm for forms with semisimple centres again using the Jordan decomposition theorem.
The authors discuss several related results and examples.
Reviewer: M. P. Chaudhary (New Delhi)Centers of multilinear forms and applicationshttps://zbmath.org/1517.150182023-09-22T14:21:46.120933Z"Huang, Hua-Lin"https://zbmath.org/authors/?q=ai:huang.hua-lin"Lu, Huajun"https://zbmath.org/authors/?q=ai:lu.huajun"Ye, Yu"https://zbmath.org/authors/?q=ai:ye.yu"Zhang, Chi"https://zbmath.org/authors/?q=ai:zhang.chi.4Summary: The center algebra of a general multilinear form is defined and investigated. We show that the center of a nondegenerate multilinear form is a finite dimensional commutative algebra, and center algebras can be effectively applied to direct sum decompositions of multilinear forms. As an application of the algebraic structure of centers, we show that almost all multilinear forms are absolutely indecomposable. The theory of centers can also be applied to symmetric equivalence of multilinear forms. Moreover, with a help of the results of symmetric equivalence, we are able to provide a linear algebraic proof for a well known Torelli type result which says that two complex homogeneous polynomials with the same Jacobian ideal are linearly equivalent.A survey on rational curves on complex surfaceshttps://zbmath.org/1517.320392023-09-22T14:21:46.120933Z"Barbaro, Giuseppe"https://zbmath.org/authors/?q=ai:barbaro.giuseppe"Fagioli, Filippo"https://zbmath.org/authors/?q=ai:fagioli.filippo"Ríos Ortiz, Ángel David"https://zbmath.org/authors/?q=ai:rios-ortiz.angel-davidSummary: In this survey, we discuss the problem of the existence of rational curves on complex surfaces, both in the Kähler and non-Kähler setup. We systematically go through the Enriques-Kodaira classification of complex surfaces to highlight the different approaches applied to the study of rational curves in each class. We also provide several examples and point out some open problems.Adjoint \((1,1)\)-classes on threefoldshttps://zbmath.org/1517.320432023-09-22T14:21:46.120933Z"Höring, A."https://zbmath.org/authors/?q=ai:horing.andreasThe article under review studies the analytic analogue of the basepoint-freeness theorem: it is known that if \(X\) is a projective variety (with mild singularities) and \(D\) a divisor such that \(D-K_X\) is nef and big, then the linear system \(|mD|\) is basepoint-free if \(m\gg 1\). In particular, the latter gives rise to a morphism \(\Phi_{mD}:X\to Z\) with connected fibres and (the numerical class of) \(D\) can be written \(D=\Phi^*(A)\) where \(A\) is an ample \(\mathbb{Q}\)-divisor on \(Z\). If \(D\) is replaced with a \((1,1\))-class, it makes perfectly sense to consider the following conjecture [\textit{S. Filip} and \textit{V. Tosatti}, Ann. Inst. Fourier 68, No. 7, 2981--2999 (2018; Zbl 1428.14065)]:
\textbf{Conjecture.} If \(\alpha\) is a nef \((1,1)\)-class on a compact Kähler manifold \(X\) such that \(\alpha-K_X\) is big and nef, then there exists \(\Phi:X\to Z\) (with connected fibres and \(Z\) normal) such that \(\alpha=\Phi^*(\alpha_Z)\) with \(\alpha_Z\) a Kähler class on \(Z\).
It was proved in [loc. cit.] that it holds in dimension \(2\) and this article addresses the case \(\dim(X)=3\). The above conjecture holds if \(\alpha-K_X\) is assumed to be Kähler (and not only nef and big). Actually, the precise statement (Theorem 1.3) is about compact Kähler threefolds with terminal singularities.
The canonical bundle playing a central role in the satement, it is always interesting to look at the special case of \(K\)-trivial spaces. In that case, the above conjecture states that a nef and big class is automatically semi-ample (and thus the pull-back of a Kähler class by a bimeromorphic map). Proposition 1.4 settles it for terminal threefolds. In that case, the Decomposition Theorem is used to reduce the problem to the surface case.
The proof (of Theorem 1.3) is obtained by running a MMP for Kähler threefold [the author and \textit{T. Peternell}, Invent. Math. 203, No. 1, 217--264 (2016; Zbl 1337.32031)]: it ends with a meromorphic map \(X\dashrightarrow Y\), the space \(Y\) being endowed with a nef and big class \(\alpha_Y\). This class satisfies \(\dim(\mathrm{Null}(\alpha))\le 1\) and the null-locus of \(\alpha_Y\) can then be contracted to obtain the sought \(Z\).
Reviewer: Benoît Claudon (Rennes)Collapsing Calabi-Yau fibrations and uniform diameter boundshttps://zbmath.org/1517.320702023-09-22T14:21:46.120933Z"Li, Yang"https://zbmath.org/authors/?q=ai:li.yang.41|li.yang|li.yang.4|li.yang.6|li.yang.11|li.yang.2|li.yang.12|li.yang.17|li.yang.5|li.yang.7|li.yang.8|li.yang.9|li.yang.13Summary: We study Calabi-Yau metrics collapsing along a holomorphic fibration over a Riemann surface. Assuming at worst canonical singular fibres, we prove a uniform diameter bound for all fibres in the suitable rescaling. This has consequences on the geometry around the singular fibres.Relative Ding and K-stability of toric Fano manifolds in low dimensionshttps://zbmath.org/1517.320722023-09-22T14:21:46.120933Z"Nitta, Yasufumi"https://zbmath.org/authors/?q=ai:nitta.yasufumi"Saito, Shunsuke"https://zbmath.org/authors/?q=ai:saito.shunsuke"Yotsutani, Naoto"https://zbmath.org/authors/?q=ai:yotsutani.naotoSummary: The purpose of this article is to clarify all of the uniformly relatively Ding stable toric Fano threefolds and fourfolds as well as unstable ones. The key player in our classification result is the Mabuchi constants, which can be calculated by combinatorial data of the associated moment polytopes due to the work of \textit{Y. Yao} [Int. Math. Res. Not. 2022, No. 24, 19790--19853 (2022; Zbl 1510.53084)]. In this article, we give the list of uniform relative Ding stability of all toric Fano manifolds in dimension up to four with the values of the Mabuchi constants. As an application of our main theorem, we clarify the difference between relative \(K\)-stability and relative Ding stability by considering some specific toric Fano manifolds. In the proof, we used Bott tower structure of relatively Ding unstable toric Fano manifolds.Some notes on the Lê numbers in the family of line singularitieshttps://zbmath.org/1517.321002023-09-22T14:21:46.120933Z"Oleksik, Grzegorz"https://zbmath.org/authors/?q=ai:oleksik.grzegorz"Różycki, Adam"https://zbmath.org/authors/?q=ai:rozycki.adamSummary: In this paper we introduce the jumps of the Lê numbers of non-isolated singularity \(f\) in the family of line deformations. Moreover, we prove
the existence of a deformation of a non-degenerate singularity \(f\) such that the first Lê number is constant and the zeroth Lê number jumps down to zero. We also give estimations of the Lê numbers when the critical locus is one-dimensional. These give a version of the celebrated theorem of A. G. Kouchnirenko in this case.
For the entire collection see [Zbl 1509.14002].Non-Hamiltonian actions with fewer isolated fixed pointshttps://zbmath.org/1517.370592023-09-22T14:21:46.120933Z"Jang, Donghoon"https://zbmath.org/authors/?q=ai:jang.donghoon"Tolman, Susan"https://zbmath.org/authors/?q=ai:tolman.susanThe following question is known as ``McDuff conjecture'': Does there exist a non-Hamiltonian symplectic circle action on a closed, connected symplectic manifold with a non-empty discrete fixed set? \textit{S. Tolman} [Invent. Math. 210, No. 3, 877--910 (2017; Zbl 1383.53061)] answered this question by constructing a non-Hamiltonian symplectic circle action on a 6-dimensional closed, connected symplectic manifold with exactly \(32\) isolated fixed points.
In this paper the authors improve this example.
More concretely, they obtain the following main theorem:
Theorem. Given any integer \(k \geqslant 5\), there exists a non-Hamiltonian symplectic circle action on a closed, connected 6-dimensional symplectic manifold with exactly \(2k\) fixed points.
From the proof of the above theorem there stems another result:
Theorem. There exists a non-Hamiltonian symplectic circle action on a closed, connected 6-dimensional symplectic manifold \((M, \omega)\) with a generalized moment map \(\Psi: M \to \mathbb{R}/10\mathbb{Z}\simeq S^1\) satisfying the following properties: each of the level sets \(\Psi^{-1}(\pm 1)\) contains \(5\) fixed points with weights \(\pm \{2,-1,-1\}\); otherwise, the action is locally free.
The Duistermaat-Heckman function is
\[
\varphi(t)=\begin{cases} 12 -2t^2, & -1 \leqslant t \leqslant 1,\\
2 + \frac{1}{2}( t -5 )^2, & 1\leqslant t \leqslant 9. \end{cases}
\]
The reduced space \(M/_{t} S^1\) is diffeomorphic to a generalized K3 surface with \(5\) isolated \(\mathbb{Z}_2\) singularities for all \(t\in (1, 9)\), and symplectomorphic to a tame K3 surface for all \(t\in (-1, 1)\).
Finally, the authors show that any non-Hamiltonian symplectic circle action on a closed, connected 6-dimensional symplectic manifold must have at least \(10\) fixed points if it shares a number of key properties with the examples constructed in the last theorem. Indeed, one has:
Lemma. Let the circle act on a closed connected 6-dimensional symplectic manifold \((M, \omega)\) with generalized moment map \(\Psi: M\to \mathbb{R}/\mathbb{Z}\). Assume that the \(S^1\) action on \(\Psi^{-1}(0)\) is free, that the reduced space \(M//_{0} S^1\) is a K3 surface, and that every fixed point \(p\in M^{S^1}\) has weights \(\pm \{2,-1,-1\}\). Then \(M\) has at least \(10\) fixed points.
Reviewer: Jianbo Wang (Tianjin)A Landau-Ginzburg mirror theorem via matrix factorizationshttps://zbmath.org/1517.810642023-09-22T14:21:46.120933Z"He, Weiqiang"https://zbmath.org/authors/?q=ai:he.weiqiang"Polishchuk, Alexander"https://zbmath.org/authors/?q=ai:polishchuk.alexander-e"Shen, Yefeng"https://zbmath.org/authors/?q=ai:shen.yefeng"Vaintrob, Arkady"https://zbmath.org/authors/?q=ai:vaintrob.arkadySummary: For an invertible quasihomogeneous polynomial \(\boldsymbol{w}\) we prove an all-genus mirror theorem relating two cohomological field theories of Landau-Ginzburg type. On the \(B\)-side it is the Saito-Givental theory for a specific choice of a primitive form. On the \(A\)-side, it is the matrix factorization CohFT for the dual singularity \(\boldsymbol{w}^T\) with the maximal diagonal symmetry group.The \(\mathsf{CP}^{n-1}\)-model with fermions: a new lookhttps://zbmath.org/1517.810922023-09-22T14:21:46.120933Z"Bykov, Dmitri"https://zbmath.org/authors/?q=ai:bykov.dmitri-v.1|bykov.dmitri-vSummary: We elaborate the formulation of the \(\mathsf{CP}^{n-1}\) sigma model with fermions as a gauged Gross-Neveu model. This approach allows to identify the super phase space of the model as a supersymplectic quotient. Potential chiral gauge anomalies are shown to receive contributions from bosons and fermions alike and are related to properties of this phase space. Along the way we demonstrate that the worldsheet supersymmetric model is a supersymplectic quotient of a model with target space supersymmetry. Possible generalizations to other quiver supervarieties are briefly discussed.On the uniqueness of supersymmetric AdS(5) black holes with toric symmetryhttps://zbmath.org/1517.830452023-09-22T14:21:46.120933Z"Lucietti, James"https://zbmath.org/authors/?q=ai:lucietti.james"Ntokos, Praxitelis"https://zbmath.org/authors/?q=ai:ntokos.praxitelis"Ovchinnikov, Sergei G."https://zbmath.org/authors/?q=ai:ovchinnikov.sergei-gSummary: We consider the classification of supersymmetric \(\mathrm{AdS}_5\) black hole solutions to minimal gauged supergravity that admit a torus symmetry. This problem reduces to finding a class of toric Kähler metrics on the base space, which in symplectic coordinates are determined by a symplectic potential. We derive the general form of the symplectic potential near any component of the horizon or axis of symmetry, which determines its singular part for any black hole solution in this class, including possible new solutions such as black lenses and multi-black holes. We find that the most general known black hole solution in this context, found by \textit{Z.-W. Chong} et al. [Phys. Rev. Lett. 95, No. 16, Article ID 161301, 4 p. (2005; \url{doi:10.1103/PhysRevLett.95.161301})] (CCLP), is described by a remarkably simple symplectic potential. We prove that any supersymmetric and toric solution that is timelike outside a smooth horizon, with a Kähler base metric of Calabi type, must be the CCLP black hole solution or its near-horizon geometry.