Recent zbMATH articles in MSC 14Jhttps://zbmath.org/atom/cc/14J2024-09-27T17:47:02.548271ZWerkzeugBook review of: E. J. N. Looijenga, Algebraic varietieshttps://zbmath.org/1541.000072024-09-27T17:47:02.548271Z"Edixhoven, Bas"https://zbmath.org/authors/?q=ai:edixhoven.s-jReview of [Zbl 1439.14008].Book review of: C. Ciliberto, Classification of complex algebraic surfaceshttps://zbmath.org/1541.000152024-09-27T17:47:02.548271Z"Kloosterman, Remke"https://zbmath.org/authors/?q=ai:kloosterman.remke-nanneReview of [Zbl 1442.14002].Book review of: S. Kondō, \(K3\) surfaceshttps://zbmath.org/1541.000212024-09-27T17:47:02.548271Z"Peters, Chris"https://zbmath.org/authors/?q=ai:peters.chris-a-mReview of [Zbl 1453.14095].Book review of: M. Schütt and T. Shioda, Mordell-Weil latticeshttps://zbmath.org/1541.000222024-09-27T17:47:02.548271Z"Peters, Chris"https://zbmath.org/authors/?q=ai:peters.chris-a-mReview of [Zbl 1433.14002].BCOV cusp forms of lattice polarized K3 surfaceshttps://zbmath.org/1541.110442024-09-27T17:47:02.548271Z"Hosono, Shinobu"https://zbmath.org/authors/?q=ai:hosono.shinobu"Kanazawa, Atsushi"https://zbmath.org/authors/?q=ai:kanazawa.atsushiThe main objective of this work is to introduce and study a BCOV formula (named after Bershadsky, Cecotti, Ooguri and Vafa [\textit{M. Bershadsky} et al., Nucl. Phys., B 405, No. 2--3, 279--304 (1993; Zbl 0908.58074)] for the lattice polarized \(\mathrm{K}3\) surfaces.
Let \(L=U^{\oplus 3}\oplus E_8(-1)^{\oplus 2}\) be the \(\mathrm{K}3\) lattice, where \(U\) is the hyperbolic lattice of rank \(2\) and \(E_8(-1)\) is the \(E_8\) root lattice with its bilinear form being multiplied by \(-1\). Let \(M\) be a primitive sublattice of \(L\) of signature \((1,r-1)\). The orthogonal lattice \(M^\perp\) is assumed to split the hyperbolic lattice: there exists a sublattice \(\check{M}\) of signature \((1,19-r)\) such that \(M^\perp=U\oplus\check{M}\). It follows from this assumption that families of \(\check{M}\)-polarized \(\mathrm{K}3\) surfaces are mirror symmetric to families of \(M\)-polarized \(\mathrm{K}3\) surfaces. The authors define the BCOV formula for the set of \(\check{M}\)-polarizable \(\mathrm{K}3\) surfaces (meaning that there exists \(\phi\colon H^2(X,\mathbb{Z})\to L\) such that \((X,\phi)\) is a \(\check{M}\)-polarized \(\mathrm{K}3\) surface) whose isomorphism classes are parametrized by the quotient of the period domain of \(\check{M}^\perp\) by the subgroup of the orthogonal group of \(\check{M}^\perp\) which fixes the period domain of \(\check{M}^\perp\).
The authors study more deeply the case of families of \(\check{M}_{2n}\)-polarizable \(\mathrm{K}3\) surfaces over \(\mathbb{P}_1\) if \(M_{2n}\) is \(\langle 2n\rangle\). In this case, the inverse a the BCOV formula is given by a cusp form explicitly described as a product of \(\eta\) functions and they conjecture that their result can be extended to general families of \(\check{M}_{2n}\)-polarizable \(\mathrm{K}3\) surfaces.
They also consider the case where \(\check{M}\) is \(U\oplus E_8(-1)\oplus E_7(-1)\) and prove that the inverse of the BCOV formula is described by the (genus \(2\)) Igusa cuspforms of weight \(10\) and \(12\) and the genus two Eisenstein series of weight \(4\).
Finally, they give the \(\mathrm{K}3\) differential operators for the \(65\) genus \(0\) groups obtained by adjoining to \(\Gamma_0(n)\) the Atkin-Lehner involutions \(W_e\) for all divisors \(e\) of \(n\) such that \(n/e\) is coprime with \(e\).
Reviewer: Emmanuel Royer (Aubière)Some moduli spaces of 1-dimensional sheaves on an elliptic ruled surfacehttps://zbmath.org/1541.140162024-09-27T17:47:02.548271Z"Yoshioka, Kōta"https://zbmath.org/authors/?q=ai:yoshioka.kotaSummary: We shall study moduli spaces of stable 1-dimensional sheaves on an elliptic ruled surface. In particular we shall compute the generating series of Hodge numbers of some moduli spaces of stable 1-dimensional sheaves.Some Betti numbers of the moduli of 1-dimensional sheaves on \(\mathbb{P}^2\)https://zbmath.org/1541.140172024-09-27T17:47:02.548271Z"Yuan, Yao"https://zbmath.org/authors/?q=ai:yuan.yaoSummary: Let \(M(d,\chi)\), with \((d,\chi)=1\), be the moduli space of semistable sheaves on \(\mathbb{P}^2\) supported on curves of degree \(d\) and with Euler characteristic \(\chi\). The cohomology ring \(H^*(M(d,\chi),\mathbb{Z})\) of \(M(d,\chi)\) is isomorphic to its Chow ring \(A^*(M(d,\chi))\) by
\textit{E. Markman}'s result [Adv. Math. 208, No. 2, 622--646 (2007; Zbl 1115.14036)].
\textit{W. Pi} and \textit{J. Shen} [Algebr. Geom. 10, No. 4, 504--520 (2023; Zbl 1528.14016)]
have described a minimal generating set of \(A^*(M(d,\chi))\) consisting of \(3d-7\) generators, which they also showed to have no relation in \(A^{\leq d-2}(M(d,\chi))\). We compute the two Betti numbers \(b_{2(d-1)}\) and \(b_{2d}\) of \(M(d,\chi)\), and as a corollary we show that the generators given by Pi and Shen have no relations in \(A^{\leq d-1}(M(d,\chi))\), but do have three linearly independent relations in \(A^d(M(d,\chi))\).Effective results in the three-dimensional minimal model programhttps://zbmath.org/1541.140252024-09-27T17:47:02.548271Z"Prokhorov, Yuri"https://zbmath.org/authors/?q=ai:prokhorov.yuri-gIn this conference report the author gives a concise survey on recent developments in the 3-dimensional minimal program (MMP for short). The MMP procedure is a sequence of Mori contractions which are constructed inductively. Recall that a contraction is a proper surjective morphism \(f:X\to Z\) of normal varieties with connected fibers. The exceptional locus of a contraction \(f\) is the subset \(Exc(f)\subset X\) of points at which \(f\) is not an isomorphism. A Mori contraction is a contraction \(f:X\to Z\) such that the variety \(X\) has at worst terminal \(\mathbb Q\)-factorial singularities, the anticanonical class \(-K_X\) is \(f\)-ample, and the relative Picard number \(\rho(X/Z)\) equals 1.
The aim of the MMP is to find a good representative in a fixed birational equivalence class of algebraic varieties. Starting with an arbitrary smooth projective variety, one can perform a finite number of certain Mori contractions and at the end obtain a variety which is simpler in some sense. The MMP has a number of applications in algebraic geometry. The most impressive consequence of the MMP is the finite generation of the canonical ring of a smooth projective variety \(X\). Another application of the MMP is the so-called Sarkisov program which allows decomposing birational maps between Mori fiber spaces into composition of elementary transformations, called Sarkisov links. The MMP can also be applied to varieties with finite group actions and to varieties over nonclosed fields.
For the entire collection see [Zbl 1532.00037].
Reviewer: Ivo M. Michailov (Shumen)Fano Shimura varieties with mostly branched cuspshttps://zbmath.org/1541.140412024-09-27T17:47:02.548271Z"Maeda, Yota"https://zbmath.org/authors/?q=ai:maeda.yota"Odaka, Yuji"https://zbmath.org/authors/?q=ai:odaka.yujiSummary: We prove that the Satake-Baily-Borel compactification of certain Shimura varieties are Fano varieties, Calabi-Yau varieties or have ample canonical divisors with mild singularities. We also prove some variants statements, give applications and discuss various examples including new ones, for instance, the moduli spaces of unpolarized (log) Enriques surfaces.
For the entire collection see [Zbl 1515.14010].Number of triple points on complete intersection Calabi-Yau threefoldshttps://zbmath.org/1541.140532024-09-27T17:47:02.548271Z"Grzelakowski, Kacper"https://zbmath.org/authors/?q=ai:grzelakowski.kacperThe authors provides examples of complete intersection Calabi-Yau threefolds with ordinary triple points and discusses upper bounds for the maximal number \(\mu_3\) of such points. The arguments differ case by case.
By projection of the complete intersection \(X_{2,4}\) of type \((2,4)\) from a triple point a normal quintic hypersurface with one triple point less and 24 double points is obtained. The spectral bound gives \(\mu_3(X_{2,4})\leq 10\). An \(X_{2,4}\) with 7 singular points in general position is constructed.
An \(X_{3,3}\) with 9 triple points is constructed with the property that for each triple point there is a cubic hypersurface which has multiplicity 3 in the point and contains the \(X_{3,3}\). With this property 9 is shown to be maximal.
For \(X_{2,2,3}\) an easy example with 4 triple points is given, while it is shown that \(X_{2,2,2,2}\) cannot contain triple points. The same is shown for the hypersurfaces \(X_8\subset \mathbb P(1 : 1 : 1 : 1 : 4)\) or \(X_{10}\subset P(1 : 1 : 1 : 2 : 5)\) in the indicated weighted projective spaces.
For \(X_6\subset \mathbb P(1 : 1 : 1:1 : 2)\) the singular points have to lie on a sextic surface that is a hyperplane section of X, and they are ordinary triple points of that surface. Such a surface can have at most ten triple points, and the triple cover of the hyperplane branched over the surface gives an example of that \(\mu_3(X_6)=10\).
Reviewer: Jan Stevens (Göteborg)On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlovhttps://zbmath.org/1541.140542024-09-27T17:47:02.548271Z"Bai, Shaoyun"https://zbmath.org/authors/?q=ai:bai.shaoyun"Côté, Laurent"https://zbmath.org/authors/?q=ai:cote.laurentThis paper studies the Rouquier dimension of the (derived) wrapped Fukaya category \(\mathcal{W} (X,A)\) associated to a Weinstein manifold \(X\) and a Weinstein hypersurface \(A\subseteq X\) in its ideal boundary. In the presence of a global Lagrangian distribution \(\xi\) on \(X\), the authors provide an upper bound for the Rouquier dimension \(\mathrm{Rdim}\ \mathcal{W} (X,A)\) of the wrapped Fukaya category, which, when homological mirror symmetry is proved, completely determines the Rouquier dimension of \(\mathcal{W}(X,A)\) in dimensions \(\leq 3\) and resolves a conjecture of Orlov that \(D^b\mathrm{Coh} (Y)\), the derived category of coherent sheaves on the mirror variety \(Y\), has Rouquier dimension equal to its dimension if \(\dim Y\leq 3\).
The authors also discuss applications of this numerical upper bound to quantitative questions in symplectic topology. They show that the minimal number of critical points \(\mathrm{Lef}_w (X)\) of possible Lefschetz fibrations on \(X\) and the intersection numbers \(\vert\mathfrak{c}_X\cap\phi (\mathfrak{c}_X)\vert \) of a core of \(X\) with its image under a generic compactly supported Hamiltonian diffeomorphism \(\phi\) are all bounded below by the Rouquier dimension of \(\mathcal{W} (X)\) plus \(1\), which under some algebraic assumptions is further bounded below by the Krull dimension of \(HW^{\bullet} (K,K)\) plus \(1\) as an algebra over some finite type subring \(R\subseteq SH^0 (X)\), where \(K\) is a split-generating object of \(\mathcal{W} (X)\).
The proof strategies for these two results are different. For a general upper bound on the Rouquier dimension \(\mathrm{Rdim}\ \mathcal{W} (X,A)\) of a Weinstein pair \((X,A)\) that contributes to the Orlov conjecture, the authors apply the sheaf-theoretic analogue of the local-to-global formula [\textit{S. Ganatra} et al., J. Am. Math. Soc. 37, No. 2, 499--635 (2024; Zbl 07798120)], the identification between microlocal sheaves and wrapped Fukaya categories [\textit{S. Ganatra} et al., ``Microlocal Morse theory of wrapped Fukaya categories'', Preprint, \url{arXiv:1809.08807}] and arborealization [\textit{D. Alvarez-Gavela} et al., ``Positive arborealization of polarized Weinstein manifolds'', Preprint, \url{arXiv:2011.08962}] to reduce the computation of \(\mathrm{Rdim}\ \mathcal{W} (X,A)\) to local pieces of the arboreal complex associated to the Weinstein pair \((X,A)\), whose derived categories and hence their Rouquier dimension are well-known. The local-to-global formula then provides the upper bound for any such polarized Weinstein pairs \((X,A)\).
To derive the lower bound for the Rouquier dimension, the authors prove a result similar to \textit{P. A. Bergh} et al. [Math. Z. 265, No. 4, 849--864 (2010; Zbl 1263.18006)] that under the existence of central actions of a finite-type algebra \(R\) on some triangulated category \(\mathcal{T}\), the Rouquier dimension \(\mathrm{Rdim}\ \mathcal{T}\) is bounded below by the Krull dimension \(\dim_R\mathrm{End}_{\mathcal{T}} (K,K)\) for some generator \(K\). The actions naturally appearing in wrapped Fukaya categories are closed-open maps \(SH^{\ast} (X)\to HH^{\bullet} (\mathcal{W} (X))\), and hence after finding split-generators, the authors get the lower bound.
Reviewer: Si-Yang Liu (Los Angeles)Orbifold Jacobian algebras for invertible polynomialshttps://zbmath.org/1541.140552024-09-27T17:47:02.548271Z"Basalaev, Alexey"https://zbmath.org/authors/?q=ai:basalaev.alexey"Takahashi, Atsushi"https://zbmath.org/authors/?q=ai:takahashi.atsushi.3"Werner, Elisabeth"https://zbmath.org/authors/?q=ai:werner.elisabethSummary: An important invariant of a polynomial \(f\) is its Jacobian algebra defined by its partial derivatives. Let \(f\) be invariant with respect to the action of a finite group of diagonal symmetries \(G\). We axiomatically define an orbifold Jacobian \(\mathbb{Z}/2\mathbb{Z}\)-graded algebra for the pair \((f, G)\) and show its existence and uniqueness in the case, when \(f\) is an invertible polynomial. In case when \(f\) defines an ADE singularity, we illustrate its geometric meaning.Fano 4-folds with nef tangent bundle in positive characteristichttps://zbmath.org/1541.140562024-09-27T17:47:02.548271Z"Takahashi, Yuta"https://zbmath.org/authors/?q=ai:takahashi.yuta"Watanabe, Kiwamu"https://zbmath.org/authors/?q=ai:watanabe.kiwamuFor a smooth projective variety \(X\), the positivity of its tangent bundle imposes important restrictions in its geometry. If the tangent bundle is ample then the famous Mori result (confirming a Hartshorne conjecture) states that \(X\) is a projective space. Over the complex numbers, if the tangent bundle is nef then, up to a étale cover, \(X\) is a Fano fiber space over an abelian variety, and a similar result holds in positive characteristic (see the Introduction of the paper under review and references therein). This puts the focus on the classification of Fano varieties with nef tangent bundle which, over the complex numbers, are conjectured to be homogeneous varieties (Campana-Peternell conjecture). The paper under review deals with the classification of Fano \(4\)-folds over a field of positive characteristic whose tangent is nef and whose Picard number is greater than two, extending previous results of the second author of the paper under review on lower dimensions. The main result, see Thm. 1.1, states that if \(X\) is a smoth Fano \(4\)-fold over an algebraically closed field such that its tangent bundle is nef and its Picard number is greater than one, then \(X\) is: either the product of a projective line and \({\mathbb P}^3\), or and a hyperquadric; or the product of two planes; or the product of a plane and a quadric; or of two quadrics; or the product of a line and the projectivization of the tangent to the plane; or, finally, the projectivization of a null-correlation bundle over \({\mathbb P}^3\).
Reviewer: Roberto Muñoz (Madrid)O'Grady tenfolds as moduli spaces of sheaveshttps://zbmath.org/1541.140572024-09-27T17:47:02.548271Z"Felisetti, Camilla"https://zbmath.org/authors/?q=ai:felisetti.camilla"Giovenzana, Franco"https://zbmath.org/authors/?q=ai:giovenzana.franco"Grossi, Annalisa"https://zbmath.org/authors/?q=ai:grossi.annalisaThis paper is concerned with determining when a manifold of OG10 type is birational to a moduli space of (twisted) sheaves on a \(K3\) surface. The authors obtain a numerical criterion for a manifold of OG10 type to be birational to a moduli space, and determine the Hassett divisors for which the associated Li-Pertusi-Zhao (LPZ) manifold is birational to a moduli space. As an application, the authors determine a criterion for a group of birational transformations to be induced on a manifold of OG10 type by a group of automorphisms of a \(K3\) surface.
O'Grady constructed a resolution of the moduli space \(M_v(S,\theta)\) of \(\theta\)-semistable on a \(K3\) surface \(S\) with Mukai vector \(v=(2,0,-2)\) and \(\theta\) a primitive \(v\)-generic polarization, which is \textit{irreducible holomorphic symplectic} (ihs) of dimension \(10\). A manifold of \textit{OG10 type} is an ihs manifold which is deformation equivalent to the resolution of \(M_v(S,\theta)\). Perego and Rapagnetta obtained a resolution resolution \(\widetilde{M}_v(S,\theta)\), which is a manifold of OG10 type, of the moduli space \(M_v(S,\theta)\) where \(v=2w\) and \(w=(r,l,s)\) is a primitive Mukai vector such that \(w^2=2\), \(r\geq 0,l\in NS(S)\) and \(l\) is the first Chern class of a line bundle in case \(r=0.\)
Kuznetsov proved that the bounded derived category of a smooth cubic fourfold \(Y\) admits a semiorthogonal decomposition of the form \(D^b(Y)=\langle \mathcal{A}(Y),\mathcal{O}_Y,\mathcal{O}_Y(1),\mathcal{O}_Y(2)\rangle.\) The algebraic Mukai lattice of \(\mathcal{A}(Y)\) always contains an \(A_2\) lattice spanned by the classes \(\lambda_i:=\Pr[\mathcal{O}_L(i)]\) where \(L\subset Y\) is a line and \(\Pr:D^b(Y)\rightarrow \mathcal{A}(Y)\) is the natural projection functor. Li, Pertusi and Zhao provided a Bridgeland stability condition \(\sigma\) for which the moduli space \(X_Y:=M_\sigma(2(\lambda_1+\lambda_2),\mathcal{A}(Y))\) admits a resoluton \(\widetilde{X}_Y\) of OG10 type. The manifold \(\widetilde{X}_Y\) is called the \textit{Li-Pertusi-Zhao} (LPZ) manifold associated to \(Y\).
In Section 3, the authors define a \textit{numerical moduli space} as an ihs manifold \(X\) such that there is a primitive class \(\sigma\in H^{1,1}(X,\mathbb{Z})\) of divisibility \(3\) and \(\sigma^2=-6\) such that \(\sigma^{\perp}\) embeds a copy of the hyperbolic plane \(U\) in \(\Lambda^{1,1}_{24}\) via the unique embedding in the even unimodular lattice \(\Lambda_{24}\) of rank \(24\) with orthogonal complement of type \((1,1)\). The main result of the section is that an ihs manifold \(X\) of OG10 type is birational to \(\widetilde{M}_v(S,\theta)\) if and only if \(X\) is a numerical moduli space.
In Section 4, the authors generalize the result of Section 3 to the twisted case. The authors define a \textit{twisted numerical moduli space} in the same way as a numerical moduli space, where the hyperbolic plane can be replaced by the twisted one \(U(n)\) for \(n\in\mathbb{N}_{>0}\). They prove that an ihs manifold \(X\) of OG10 type is birational to the desingularization of the moduli space of twisted sheaves \(\widetilde{M}_v(S,\alpha,\theta)\), where \(S\) is a \(K3\) surface and \(\alpha\in Br(S)=H^2(S,\mathcal{O}_S^*)_{tor}\), if and only if \(X\) is a twisted numerical moduli space.
In Section 5, the authors give a criterion for determining when the LPZ manifold \(\widetilde{X}_Y\) associated to a cubic fourfold \(Y\) is birational to a (twisted) moduli space of sheaves on a \(K3\) surface. The irreducible Hassett divisor \(\mathcal{C}_d\) consists of cubic fourfolds \(Y\) having a primitive rank \(2\) lattice \(K\subset H^4(Y,\mathbb{Z})\) containing the square of a hyperplane class, such that \(K\) has discriminant \(d\). The authors prove that \(\widetilde{X}_Y\) is brational to a desingularized moduli space \(\widetilde{M}_v(S,\theta)\) if and only if \(Y\in\mathcal{C}_d\) such that \(d\) divides \(2n^2+2n+2\) for some \(n\in\mathbb{Z}\). They also prove that \(\widetilde{X}_Y\) is brational to a desingualrized twisted moduli space \(\widetilde{M}_v(S,\alpha,\theta)\) if and only if \(Y\in\mathcal{C}_d\) such that in the prime factorization \(\frac{d}{2}\), primes \(p\equiv 2(3)\) appear with even exponents.
In section 6, the authors prove that a group \(G\subset Bir(X)\) of birational transformations of a manifold \(X\) of OG10 type is induced by a group of automorphisms of the \(K3\) surface such that \(X\) is birational to \(\widetilde{M}_v(S,\theta)\) if and only if \(G\) is \textit{numerically induced}, i.e. \(X\) is a numerical moduli space and the class \(\sigma\) and the hyperbolic plane \(U\) are pointwise fixed by the action of \(G\). As a corollary, the authors determine that a birational symplectic involution \(\phi\in Bir(X)\) is induced if and only if the associated coinvarian lattice \(H^2(X,\mathbb{Z})_\phi\) is isometric to the lattice \(E_8(-2)\).
Reviewer: Simone Billi (Bologna)Perverse-Hodge complexes for Lagrangian fibrationshttps://zbmath.org/1541.140582024-09-27T17:47:02.548271Z"Shen, Junliang"https://zbmath.org/authors/?q=ai:shen.junliang"Yin, Qizheng"https://zbmath.org/authors/?q=ai:yin.qizhengSummary: Perverse-Hodge complexes are objects in the derived category of coherent sheaves obtained from Hodge modules associated with Saito's decomposition theorem. We study perverse-Hodge complexes for Lagrangian fibrations and propose a symmetry between them. This conjectural symmetry categorifies the authors' ``perverse = Hodge'' identity and specializes to Matsushita's theorem on the higher direct images of the structure sheaf. We verify our conjecture in several cases by making connections with variations of Hodge structures, Hilbert schemes, and Looijenga-Lunts-Verbitsky Lie algebras.Birational rigidity and K-stability of Fano hypersurfaces with ordinary double pointshttps://zbmath.org/1541.140592024-09-27T17:47:02.548271Z"de Fernex, Tommaso"https://zbmath.org/authors/?q=ai:de-fernex.tommasoA projective variety of dimension \(n\) is rational when it is birational to the projective space \(\mathbb{P}^n\). Excluding obvious cases, establishing rationality of a projective variety is a daunting task to which many authors gave their contribution through the years.
Relatively recently, the case of smooth hypersurfaces of degree \(n+1\) in \(\mathbb{P}^{n+1}\) (for \(n\geq 3\)) was settled. Namely, it was proved that these varieties are birationally superrigid, hence not rational (main references are: [\textit{V. A. Iskovskikh} and \textit{Yu. I. Manin}, Mat. Sb., Nov. Ser. 86(128), 140--166 (1971; Zbl 0222.14009); \textit{A. V. Pukhlikov}, Invent. Math. 87, 303--329 (1987; Zbl 0613.14011); \textit{A. V. Pukhlikov}, Invent. Math. 134, No. 2, 401--426 (1998; Zbl 0964.14011); \textit{T. de Fernex}, Invent. Math. 203, No. 2, 675--680 (2016; Zbl 1441.14045)].
In the paper under review, the previous result is extended to singular hypersurfaces. The author proves that for \(n\geq 5\) all hypersurfaces of degree \(n+1\) in \(\mathbb{P}^{n+1}\) with isolated ordinary double points are birationally superrigid (hence non rational) and \(K\)-stable (hence admit a weak Kähler-Einstein metric).
Reviewer: Davide Fusi (Bluffton)On K-semistable domains -- more exampleshttps://zbmath.org/1541.140602024-09-27T17:47:02.548271Z"Zhou, Chuyu"https://zbmath.org/authors/?q=ai:zhou.chuyuK-stability has arguably become one of the most important properties for Fano avrieties as it is equivalent to existence of a Kähler-Einstein metric and it also provides the right platform for constructing compact moduli spaces for Fano avrieties. A fruitful technique, notably in moduli studyies, is to look at K-stability of pairs \((X,\alpha D)\), and varrying \(\alpha\) to detect an interval in which the pair remains K-semistable. In other work, the author has generalised this type of study to pairs with several boundaries, i.e., \((X,\alpha_1 D_1+\cdots+\alpha_mD_m)\). In this paper, the author provides several concrete cases for that theory and produces some technical, yet fruitful, families of examples in which K-semistable domain is precisely computed.
Reviewer: Hamid Abban (Nottingham)Homogeneous ACM bundles on Grassmannians of exceptional typeshttps://zbmath.org/1541.140612024-09-27T17:47:02.548271Z"Fang, Xinyi"https://zbmath.org/authors/?q=ai:fang.xinyi"Nakayama, Yusuke"https://zbmath.org/authors/?q=ai:nakayama.yusuke"Ren, Peng"https://zbmath.org/authors/?q=ai:ren.pengThe paper under review presents a contribution to the study of vector bundles over projective varieties, particularly on Grassmannians of exceptional types.
The authors' motivation is to classify all irreducible homogeneous ACM bundles over generalized Grassmannians (rational homogeneous varieties with Picard group \(\mathbb{Z}\)), which is a generalization of previous work on usual Grassmannians [\textit{L. Costa} and \textit{R. M. Miró-Roig}, Adv. Math. 289, 95--113 (2016; Zbl 1421.14006)] and isotropic Grassmannians [\textit{R. Du} et al., Forum Math. 35, No. 3, 763--782 (2023; Zbl 1524.14093)]. This is an interesting problem in algebraic geometry as ACM bundles have various applications and connections to other areas of mathematics.
The main result of the paper, which is based on the Borel-Bott-Weil theorem, provides a necessary and sufficient condition for an irreducible homogeneous vector bundle to be an ACM bundle in terms of its associated datum. This theorem can be used to characterize ACM bundles and has interesting consequences, such as Corollary 1.2, which states that only finitely many irreducible homogeneous ACM bundles up to tensoring a line bundle exist on a given generalized Grassmannian.
The paper also presents a detailed analysis of the concrete form of the associated datum on generalized Grassmannians of different types, especially on Grassmannians of exceptional types. This analysis is carried out through explicit calculations of positive roots and the use of the Killing form, which leads to concrete examples and classifications of ACM bundles in these cases.
Furthermore, the authors determine the representation type of some Grassmannians of exceptional types. By constructing families of indecomposable ACM bundles, they show that certain Grassmannians are of wild representation type.
Reviewer: Rong Du (Shanghai)Projective rigidity and Alexander polynomials of certain nodal hypersurfaceshttps://zbmath.org/1541.140622024-09-27T17:47:02.548271Z"Escudero, Juan García"https://zbmath.org/authors/?q=ai:garcia-escudero.juanSummary: We present nodal algebraic hypersurfaces in the complex projective space which are projectively rigid. Defects and Alexander polynomials associated with the hypersurfaces are obtained. There are families of nodal hypersurfaces with nontrivial Alexander polynomials and nodal threefolds with projective rigidity which are potentially infinite.The ice cone family and iterated integrals for Calabi-Yau varietieshttps://zbmath.org/1541.140632024-09-27T17:47:02.548271Z"Duhr, Claude"https://zbmath.org/authors/?q=ai:duhr.claude"Klemm, Albrecht"https://zbmath.org/authors/?q=ai:klemm.albrecht"Nega, Christoph"https://zbmath.org/authors/?q=ai:nega.christoph"Tancredi, Lorenzo"https://zbmath.org/authors/?q=ai:tancredi.lorenzoSummary: We present for the first time fully analytic results for multi-loop equal-mass ice cone graphs in two dimensions. By analysing the leading singularities of these integrals, we find that the maximal cuts in two dimensions can be organised into two copies of the same periods that describe the Calabi-Yau varieties for the equal-mass banana integrals. We obtain a conjectural basis of master integrals at an arbitrary number of loops, and we solve the system of differential equations satisfied by the master integrals in terms of the same class of iterated integrals that have appeared earlier in the context of equal-mass banana integrals. We then go on and show that, when expressed in terms of the canonical coordinate on the moduli space, our results can naturally be written as iterated integrals involving the geometrical invariants of the Calabi-Yau varieties. Our results indicate how the concept of pure functions and transcendental weight can be extended to the case of Calabi-Yau varieties. Finally, we also obtain a novel representation of the periods of the Calabi-Yau varieties in terms of the same class of iterated integrals, and we show that the well-known quadratic relations among the periods reduce to simple shuffle relations among these iterated integrals.Projective varieties with nef tangent bundle in positive characteristichttps://zbmath.org/1541.140692024-09-27T17:47:02.548271Z"Kanemitsu, Akihiro"https://zbmath.org/authors/?q=ai:kanemitsu.akihiro"Watanabe, Kiwamu"https://zbmath.org/authors/?q=ai:watanabe.kiwamuThe setting is that of smooth projective varieties \(X\) (over an algebraically closed field) whose tangent bundle satisfies the numerical semipositivity condition of being \emph{nef}. Then \(X\) is expected to decompose into a ``positive'' and a ``flat'' part; over the complex numbers, this has been proven by Demailly, Peternell and Schneider using analytical methods.
The aim of this paper is to establish the analogous decomposition result in the characteristic \(p >0\) case. The strategy is to use Mori theory: the first result (Theorem 1.3) asserts that a rationally chain connected \(X\) is actually separably rationally connected (SRC). The second key point is the existence, for a given \(K_X\)-negative extremal ray, of a smooth contraction whose fibers are SRC Fano varieties, and whose fibers and target both still have nef tangent bundle.
This yields to the main result of the paper (Theorem 1.8, decomposition theorem): \(X\) admits a smooth contraction, whose fibers are smooth SRC Fano varieties with nef tangent bundle, and whose target has numerically trivial tangent bundle.
Thus, the study of varieties with nef tangent bundle is reduced to the Fano case and the \(K_X\)-trivial case.
As an application, the authors consider \(F\)-liftable varieties, namely smooth projective varieties which lift modulo \(p^2\) with respect to the Frobenius morphism. The following is proven (Theorem 1.12): if \(X\) is \(F\)-liftable and has nef tangent bundle, then it can be realized as a finite étale quotient of some \(Y\), such that the Albanese map of \(Y\) is a smooth morphism onto an ordinary Abelian variety, whose fibers are product of projective spaces.
Reviewer: Matilde Maccan (Rennes)Rouquier dimension is Krull dimension for normal toric varietieshttps://zbmath.org/1541.140702024-09-27T17:47:02.548271Z"Favero, David"https://zbmath.org/authors/?q=ai:favero.david"Huang, Jesse"https://zbmath.org/authors/?q=ai:huang.jesseSummary: We prove that for any normal toric variety, the Rouquier dimension of its bounded derived category of coherent sheaves is equal to its Krull dimension. Our proof uses the coherent-constructible correspondence to translate the problem into the study of Rouquier dimension for certain categories of constructible sheaves.Canonical Kähler metrics and stability of algebraic varietieshttps://zbmath.org/1541.320072024-09-27T17:47:02.548271Z"Li, Chi"https://zbmath.org/authors/?q=ai:li.chiThis is a nice and brief survey of the theory of canonical metrics on algebraic varieties and their existence as characterized by K-stability.
It goes back to Calabi to find canonical Riemannian metrics on algebraic varieties, as a vast generalization of the uniformization theorem. For projective varieties with either trivial or ample canonical bundle, \textit{S.-T. Yau} [Commun. Pure Appl. Math. 31, 339--411 (1978; Zbl 0369.53059)] and independently \textit{T. Aubin} [Bull. Sci. Math., II. Sér. 102, 63--95 (1978; Zbl 0374.53022)] proved the existence of Kähler-Einstein metrics with vanishing and negative Ricci curvature, respectively. In the case that the anti-canonical line bundle is ample, there are obstructions to existence of a positively curved Kähler-Einstein metric going back to \textit{Y. Matsushima} in [Nagoya Math. J. 11, 145--150 (1957; Zbl 0091.34803)]. This led to the development of an algebraic criterion (dubbed K-stability) introduced by \textit{G. Tian} [Invent. Math. 130, No. 1, 1--37 (1997; Zbl 0892.53027)], which aims to completely characterize algebraically the existence of a Kähler-Einstein metric. This conjecture, known as the Yau-Tian-Donaldsson conjecture, has by now been settled in the works [\textit{X. Chen} et al., J. Am. Math. Soc. 28, No. 1, 183--197 (2015; Zbl 1312.53096); J. Am. Math. Soc. 28, No. 1, 199--234 (2015; Zbl 1312.53097); J. Am. Math. Soc. 28, No. 1, 235--278 (2015; Zbl 1311.53059); \textit{G. Tian}, Commun. Pure Appl. Math. 68, No. 7, 1085--1156 (2015; Zbl 1318.14038)], even in the singular setting [\textit{C. Li} et al., Commun. Pure Appl. Math. 74, No. 8, 1748--1800 (2021; Zbl 1484.32041)]. On a general polarized variety, one can look for constant scalar curvature Kähler metrics (cscK), and there is a version of K-stability also in this case introduced in [\textit{S. K. Donaldson}, J. Differ. Geom. 62, No. 2, 289--349 (2002; Zbl 1074.53059)], although the polarized version of the Yau-Tian-Donaldson conjecture is yet to be confirmed.
In the first part of the survey, various canonical metrics on algebraic varieties are introduced in addition to Kähler-Einstein and cscK metrics: weighted Kähler-Ricci solitons, log Kähler-Einstein metrics and Ricci-flat cone metrics. These metrics can be realized as critical points of suitable functionals defined on the space of Kähler metrics that are introduced. In each case, the existence can be analytically characterized by a coercivity notion of the corresponding functional.
In the second part, K-stability is introduced with a strong emphasize on the viewpoint of non-Archimedean geometry following [\textit{S. Boucksom} and \textit{M. Jonsson}, ``A non-Archimedean approach to K-stability'', Preprint, \url{arXiv:1805.11160}]. The functionals from the fist part have non-Archimedean counterparts with which K-stability can be characterized. For the case of Fano varieties, there are alternative and easier-to-verify notions, such as the delta-invariant, which has been shown with tools from the minimal model program, to lead to an equivalent theory of stability.
In the third and final part, the correspondence between Kähler and non-Archimedean functionals is presented. The main idea being that the slope at infinity of various Kähler-geometric functionals along certain rays in the space of Kähler metrics is given precisely by their non-Archimedean counterparts. In the Kähler-Einstein case, this idea has been implemented to prove the Yau-Tian-Donaldsson conjecture by variational means, see [\textit{R. J. Berman} et al., J. Am. Math. Soc. 34, No. 3, 605--652 (2021; Zbl 1487.32141)]. In the cscK case, although important progress have been achieved, there are hurdles yet to be overcome.
I recommend the survey for anyone who wants a quick overview of the recent progress surrounding K-stability and canonical metrics on algebraic varieties.
For the entire collection see [Zbl 1532.00037].
Reviewer: Rolf Andreasson (Gothenburg)Translational and great Darboux cyclideshttps://zbmath.org/1541.510082024-09-27T17:47:02.548271Z"Lubbes, Niels"https://zbmath.org/authors/?q=ai:lubbes.nielsThis article characterizes real irreducible algebraic surfaces in \({\mathbb R}^3\) that contain at least two circles through each point.
Two surfaces in \({\mathbb R}^3\) are said to be Möbius equivalent if one surface is mapped to the other by a composition of inversions.
With \(\mu: {\mathbb S}^3\rightarrow {\mathbb R}^3\) the stereographic projection from the point \((0, 0, 0, 1)\) on the 3-dimensional unit-sphere \({\mathbb S}^3\subset {\mathbb R}^4\) -- \(\mu(y):= (y_1, y_2, y_3)/(1-y_4)\) -- the Möbius degree of a real irreducible algebraic surface \(Z \subset {\mathbb R}^3\) is defined as \(\deg \mu^{-1}(Z)\) and \(Z\) is called \(\lambda\)-circled if the Zariski closure of \(\mu^{-1}(Z)\) contains at least \(\lambda\in Z_{\geq 0} \cup \{\infty\}\) circles through a general point. If \(\lambda\in Z_{\geq 0}\), then by \(\lambda\)-circled one understands that \(Z\) is not \((\lambda + 1)\)-circled. If \(\lambda\geq 2\), then \(Z\) is called \textit{celestial}.
If \(A\) and \(B\) are curves in \({\mathbb R}^3\) or \({\mathbb S}^3\), one identifies the unit-sphere \({\mathbb S}^3\subset {\mathbb R}^4\) with the unit quaternions and, denoting the Hamiltonian product by \(*\) one can define, \(A + B\) and \(A*B\) in the usual manner. A real irreducible algebraic surface is said to be \textit{Bohemian} or \textit{Cliffordian} if there exist generalized circles \(A\) and \(B\) such that \(Z\) is the Zariski closure of \(A + B\) and \(\mu(A*B)\), respectively. A surface that is either Bohemian or Cliffordian is called \textit{translational}. If \(A\) and \(B\) are great circles such that \(A*B\subset {\mathbb S}^3\) is a real irreducible algebraic surface, then \(A*B\) is called a \textit{Clifford torus}. A \textit{Darboux cyclide} in \({\mathbb R}^3\) is a real irreducible algebraic surface of Möbius degree four. A \textit{\(Q\) cyclide} is a Darboux cyclide that is Möbius equivalent to a quadric \(Q\). With the following abbreviations for quadrics, \(E\) = elliptic/ellipsoid, \(P\) = parabolic/paraboloid, \(O\) = cone, \(C\) = circular, \(H\) = hyperbolic/hyperboloid, \(Y\) = cylinder, with a \textit{\(CH1\) cyclide} denoting one that is Möbius equivalent to a Circular Hyperboloid of 1 sheet, with a \textit{ring cyclide}, \textit{Perseus cyclide} or \textit{Blum cyclide} designating a Darboux cyclide without real singularities that is 4-circled, 5-circled and 6-circled, respectively, and with a real irreducible algebraic surface \(Z \subset {\mathbb R}^3\) called \textit{great} if its inverse stereographic projection \(\mu^{-1}(Z)\) is covered by great circular arcs, the main theorem states that:
For a \(\lambda\)-circled surface \(Z \subset {\mathbb R}^3\) of Möbius degree \(d\), with \(\lambda\geq 2\) and \((d, \lambda)\neq (8, 2)\), we have:
\(\bullet\) \(Z\) is Bohemian if and only if \(Z\) is either a plane, \(CY\) or \(EY\).
\(\bullet\) If \(Z\) is Cliffordian, then \(Z\) is either a Perseus cyclide, ring cyclide or \(CH1\) cyclide. Conversely, if \(Z\) is a ring cyclide, then \(Z\) is Möbius equivalent to a Cliffordian surface.
\(\bullet\) \(Z\) is Möbius equivalent to a great celestial surface if and only if \(Z\) is either a plane, sphere, Blum cyclide, Perseus cyclide, ring cyclide, \(EO\) cyclide or \(CO\) cyclide.
Reviewer: Victor V. Pambuccian (Glendale)Gravitational instantons with quadratic volume growthhttps://zbmath.org/1541.530622024-09-27T17:47:02.548271Z"Chen, Gao"https://zbmath.org/authors/?q=ai:chen.gao"Viaclovsky, Jeff"https://zbmath.org/authors/?q=ai:viaclovsky.jeff-aThis paper classifies all so-called ALG\(^*\)-type gravitational instantons using the theory of elliptic surfaces.
Recall that a gravitational instanton is a complete hyper-Kähler \(4\)-manifold \((M,g)\). This condition on the metric \(g\) implies that it is Ricci-flat (i.e., solves the Riemannian vacuum Einstein equation, hence it is called gravitational) such that its full Riemannian tensor is self-dual (hence the name instanton) with respect to the orientation on \(M\) compatible with its hyper-Kähler structure. In the case if \(M\) is non-compact various volume growth conditions on the metric combined with curvature decay assumptions are imposed too leading to their primary ALE-ALF-ALG-ALH classification. In particular if the volume has only quadratic (instead of the possible quartic) growth towards infinity then the corresponding gravitational instanton is called ALG- or ALG\(^*\)-type depending on the rate of its curvature decay. Belonging to this class intuitively means that the infinitely distant boundary of \(M\) is fibered by real \(\dim M-2=4-2=2\)-dimensional compact submanifolds which together with their metrics necessarily approach a finite volume flat geometry by the imposed volume growth and curvature decay rates; since the only flat compact surface is the torus (or the Klein bottle if it is not oriented) it is expected that the theory of ALG or ALG\(^*\) spaces is linked in a general way with the theory of elliptic surfaces, toric varieties, etc.
The results of the paper are roughly as follows:
(i) A relationship is established between (appropriately compactified) ALG\(^*\) gravitational instantons and rational elliptic surfaces (see Theorem A in the article);
(ii) This implies various rigidity results concerning their topology, i.e., how various ALG geometric data fix the diffeomorphism class of the underlying differentiable manifold (see Theorems B, C, D, E and F in the article).
Reviewer: Gábor Etesi (Budapest)Closed 3-forms in five dimensions and embedding problemshttps://zbmath.org/1541.530632024-09-27T17:47:02.548271Z"Donaldson, Simon"https://zbmath.org/authors/?q=ai:donaldson.simon-k"Lehmann, Fabian"https://zbmath.org/authors/?q=ai:lehmann.fabianThe current paper is part of a trilogy by the authors. The other two are [``Volume functionals on pseudoconvex hypersurfaces'', Preprint, \url{arXiv:2305.09932}; ``Calabi-Yau threefolds with boundary'', Preprint, \url{arXiv:2403.15184}]. These can be seen as continuations of [\textit{S. Donaldson}, Ann. Inst. Fourier 68, No. 7, 2783--2809 (2018; Zbl 1416.53024)].
In the present article, the authors ask the following question. Assume \((Z,\Psi)\) is a Calabi-Yau three-fold with chosen holomorphic volume form \(\Psi\). Let \((M,\psi)\) be a 5-manifold with a closed three-form \(\psi\). When can one find an embedding \(f\colon M\to Z\) such that \(f^*(\mathrm{Re}(\Psi))=\psi\)? The motivation comes from \textit{N. Hitchin}'s work on \(\mathrm{SL}(3,\mathbb{C})\)-structures [J. Differ. Geom. 55, No. 3, 547--576 (2000; Zbl 1036.53042)]. To answer the question, they introduce a concept of strong pseudoconvexity for three-forms and show that the perturbative version of the embedding problem can be solved if a finite-dimensional obstruction space vanishes. They also have families of examples where this obstruction space vanishes.
They attack their problem by splitting it into two parts. First, the authors construct something they call a contact hyper-Kähler \(\mathrm{SU}(2)\)-structure. This will give their 5-manifold a CR structure, and their embedding problem becomes a question of finding an embedding for a strongly pseudoconvex CR-manifold. This leads to the analysis of a sub-elliptic operator.
The paper does a good job of giving precise references when quoting technical results, especially for the analytical aspects. This should make the article more valuable both as a reference and an introduction to the topic.
Reviewer: Jørgen Olsen Lye (Hannover)Homological mirror symmetry of \(\mathbb{F}_1\) via Morse homotopyhttps://zbmath.org/1541.531062024-09-27T17:47:02.548271Z"Futaki, Masahiro"https://zbmath.org/authors/?q=ai:futaki.masahiro"Kajiura, Hiroshige"https://zbmath.org/authors/?q=ai:kajiura.hiroshigeSummary: This is a sequel to our paper [J. Math. Phys. 62, No. 3, Article ID 032307, 21 p. (2021; Zbl 1475.53095)], where we proposed a definition of the Morse homotopy of the moment polytope of toric manifolds. Using this as the substitute of the Fukaya category, we proved a version of homological mirror symmetry for the projective spaces and their products via Strominger-Yau-Zaslow construction of the mirror dual Landau-Ginzburg model.
In this paper we go this way further and extend our previous result to the case of the Hirzebruch surface \(\mathbb{F}_1\).Homological mirror symmetry of toric Fano surfaces via Morse homotopyhttps://zbmath.org/1541.531072024-09-27T17:47:02.548271Z"Nakanishi, Hayato"https://zbmath.org/authors/?q=ai:nakanishi.hayatoSummary: Strominger-Yau-Zaslow (SYZ) proposed a way of constructing mirror pairs as pairs of torus fibrations. We apply this SYZ construction to toric Fano surfaces as complex manifolds, and discuss the homological mirror symmetry, where we consider Morse homotopy of the moment polytope instead of the Fukaya category.
{\copyright 2024 American Institute of Physics}Ungraded matrix factorizations as mirrors of non-orientable Lagrangianshttps://zbmath.org/1541.531082024-09-27T17:47:02.548271Z"Amorim, Lino"https://zbmath.org/authors/?q=ai:amorim.lino"Cho, Cheol-Hyun"https://zbmath.org/authors/?q=ai:cho.cheol-hyunSummary: We introduce the notion of ungraded matrix factorization as a mirror of non-orientable Lagrangian submanifolds. An ungraded matrix factorization of a polynomial \(W\), with coefficients in a field of characteristic 2, is a square matrix \(Q\) of polynomial entries satisfying \(Q^2 = W \cdot \operatorname{Id}\). We then show that non-orientable Lagrangians correspond to ungraded matrix factorizations under the localized mirror functor and illustrate this construction in a few instances. Our main example is the Lagrangian submanifold \(\mathbb{R} P^2 \subset \mathbb{C} P^2\) and its mirror ungraded matrix factorization, which we construct and study. In particular, we prove a version of Homological Mirror Symmetry in this setting.On the lower boundedness of modified \(K\)-energyhttps://zbmath.org/1541.531222024-09-27T17:47:02.548271Z"Zhang, Liang"https://zbmath.org/authors/?q=ai:zhang.liang.14|zhang.liang.2|zhang.liang.3|zhang.liang|zhang.liang.1|zhang.liang.7|zhang.liang.5Summary: In this paper we prove that the modified \(K\)-energy on a Fano manifold \(X\) is bounded from below if \(X\) admits a special degeneration whose central fiber is a Kähler-Ricci soliton and the soliton vector field can be lifted to \(X\). We will use this result to derive some equivalent criteria of the modified \(K\)-semistability. We will also use this result to study the limit of Kähler-Ricci flow.Topological strings on non-commutative resolutionshttps://zbmath.org/1541.810952024-09-27T17:47:02.548271Z"Katz, Sheldon"https://zbmath.org/authors/?q=ai:katz.sheldon"Klemm, Albrecht"https://zbmath.org/authors/?q=ai:klemm.albrecht"Schimannek, Thorsten"https://zbmath.org/authors/?q=ai:schimannek.thorsten"Sharpe, Eric"https://zbmath.org/authors/?q=ai:sharpe.eric-rSummary: In this paper we propose a definition of torsion refined Gopakumar-Vafa (GV) invariants for Calabi-Yau threefolds with terminal nodal singularities that do not admit Kähler crepant resolutions. Physically, the refinement takes into account the charge of five-dimensional BPS states under a discrete gauge symmetry in M-theory. We propose a mathematical definition of the invariants in terms of the geometry of all non-Kähler crepant resolutions taken together. The invariants are encoded in the A-model topological string partition functions associated to non-commutative (nc) resolutions of the Calabi-Yau. Our main example will be a singular degeneration of the generic Calabi-Yau double cover of \({\mathbb{P}}^3\) and leads to an enumerative interpretation of the topological string partition function of a hybrid Landau-Ginzburg model. Our results generalize a recent physical proposal made in the context of torus fibered Calabi-Yau manifolds by one of the authors and clarify the associated enumerative geometry.Unifying the 6D \(\mathcal{N} = (1, 1)\) string landscapehttps://zbmath.org/1541.811322024-09-27T17:47:02.548271Z"Fraiman, Bernardo"https://zbmath.org/authors/?q=ai:fraiman.bernardo"De Freitas, Héctor Parra"https://zbmath.org/authors/?q=ai:de-freitas.hector-parraSummary: We propose an organizing principle for string theory moduli spaces in six dimensions with \(\mathcal{N} = (1, 1)\), based on a rank reduction map, into which all known constructions fit. In the case of cyclic orbifolds, which are the main focus of the paper, we make an explicit connection with meromorphic 2D (s)CFTs with \(c = 24\) (\(c = 12\)) and show how these encode every possible gauge symmetry enhancement in their associated 6D theories. These results generalize naturally to non-cyclic orbifolds, into which the only known string construction (to our awareness) also fits. This framework suggests the existence of a total of 47 moduli spaces: the Narain moduli space, 23 of cyclic orbifold type and 23 of non-cyclic type. Of these only 17 have known string constructions. Among the 30 new moduli spaces, 15 correspond to pure supergravity, for a total of 16 such spaces. A full classification of nonabelian gauge symmetries is given, and as a byproduct we complete the one for seven dimensions, in which only those of theories with heterotic descriptions were known exhaustively.Symmetries of Calabi-Yau prepotentials with isomorphic flopshttps://zbmath.org/1541.811372024-09-27T17:47:02.548271Z"Lukas, Andre"https://zbmath.org/authors/?q=ai:lukas.andre"Ruehle, Fabian"https://zbmath.org/authors/?q=ai:ruehle.fabianSummary: Calabi-Yau threefolds with infinitely many flops to isomorphic manifolds have an extended Kähler cone made up from an infinite number of individual Kähler cones. These cones are related by reflection symmetries across flop walls. We study the implications of this cone structure for mirror symmetry, by considering the instanton part of the prepotential in Calabi-Yau threefolds. We show that such isomorphic flops across facets of the Kähler cone boundary give rise to symmetry groups isomorphic to Coxeter groups. In the dual Mori cone, non-flopping curve classes that are identified under these groups have the same Gopakumar-Vafa invariants. This leads to instanton prepotentials invariant under Coxeter groups, which we make manifest by introducing appropriate invariant functions. For some cases, these functions can be expressed in terms of theta functions whose appearance can be linked to an elliptic fibration structure of the Calabi-Yau manifold.Interior analysis, stretched technique and bubbling geometrieshttps://zbmath.org/1541.811462024-09-27T17:47:02.548271Z"Jia, Qiuye"https://zbmath.org/authors/?q=ai:jia.qiuye"Lin, Hai"https://zbmath.org/authors/?q=ai:lin.hai.1Summary: We perform a detailed analysis of quarter BPS bubbling geometries with AdS asymptotics and their corresponding duality relations with their dual states in the quantum field theory side, among other aspects. We derive generalized Laplace-type equations with sources, obtained from linearized Monge-Ampere equations, and used for asymptotically AdS geometry. This enables us to obtain solutions specific to the asymptotically AdS context. We conduct a thorough analysis of boundary conditions and explore the stretched technique where boundary conditions are imposed on a stretched surface. These boundary conditions include grey droplets. This stretched technique is naturally used for the superstar, where we place grey droplet boundary conditions on the stretched surface. We also perform a coarse-graining of configurations and analyze the symplectic forms on the configuration space and their coarse-graining.Heavy axions from twin dark sectors with \(\bar{\theta}\)-characterized mirror symmetryhttps://zbmath.org/1541.812182024-09-27T17:47:02.548271Z"Gu, Pei-Hong"https://zbmath.org/authors/?q=ai:gu.pei-hongSummary: The QCD Lagrangian contains a CP violating gluon density term with a physical coefficient \(\bar{\theta}\). The upper bound on the electric dipole moment of neutron requires the value of \(\bar{\theta}\) to be extremely small. The tiny \(\bar{\theta}\) is commonly known as the strong CP problem. In order to solve this puzzle, we construct a \(\bar{\theta}\)-characterized mirror symmetry between a pair of twin dark sectors with respective discrete symmetries. By taking a proper phase rotation of dark fields, we can perfectly remove the parameter \(\bar{\theta}\) from the full Lagrangian. In our scenario, the discrete symmetry breaking, which are responsible for the mass generation of dark colored fermions and dark matter fermions, can be allowed near the TeV scale. This means different phenomena from the popular axion models with high scale Peccei-Quinn global symmetry breaking.