Recent zbMATH articles in MSC 14Jhttps://www.zbmath.org/atom/cc/14J2021-04-16T16:22:00+00:00WerkzeugThe pluricanonical systems of a product-quotient variety.https://www.zbmath.org/1456.140582021-04-16T16:22:00+00:00"Favale, Filippo F."https://www.zbmath.org/authors/?q=ai:favale.filippo-francesco"Gleissner, Christian"https://www.zbmath.org/authors/?q=ai:gleissner.christian"Pignatelli, Roberto"https://www.zbmath.org/authors/?q=ai:pignatelli.robertoSummary: We give a method for the computation of the plurigenera of a product-quotient manifold, and two different types of applications of it: to the construction of Calabi-Yau threefolds and to the determination of the minimal model of a product-quotient surface of general type.
For the entire collection see [Zbl 07237934].Systematics of type IIA moduli stabilisation.https://www.zbmath.org/1456.831052021-04-16T16:22:00+00:00"Marchesano, Fernando"https://www.zbmath.org/authors/?q=ai:marchesano.fernando"Prieto, David"https://www.zbmath.org/authors/?q=ai:prieto.david"Quirant, Joan"https://www.zbmath.org/authors/?q=ai:quirant.joan"Shukla, Pramod"https://www.zbmath.org/authors/?q=ai:shukla.pramod-sSummary: We analyse the flux-induced scalar potential for type IIA orientifolds in the presence of \(p\)-form, geometric and non-geometric fluxes. Just like in the Calabi-Yau case, the potential presents a bilinear structure, with a factorised dependence on axions and saxions. This feature allows one to perform a systematic search for vacua, which we implement for the case of geometric backgrounds. Guided by stability criteria, we consider configurations with a particular on-shell F-term pattern, and show that no de Sitter extrema are allowed for them. We classify branches of supersymmetric and non-supersymmetric vacua, and argue that the latter are perturbatively stable for a large subset of them. Our solutions reproduce and generalise previous results in the literature, obtained either from the 4d or 10d viewpoint.Del Pezzo surfaces with infinite automorphism groups.https://www.zbmath.org/1456.140512021-04-16T16:22:00+00:00"Cheltsov, Ivan"https://www.zbmath.org/authors/?q=ai:cheltsov.ivan"Prokhorov, Yuri"https://www.zbmath.org/authors/?q=ai:prokhorov.yuri-gIn this interesting and nicely written paper the authors provide a complete classification of del Pezzo surfaces \(X\) defined over an algebraically closed field of characteristic zero with at worst Du Val singularities such that their automorphism group \(\mathrm{Aut}(X)\) is infinite (see Theorem 1.1 therein). They also classify all those Du Val del Pezzo surfaces for which the automorphism group \(\mathrm{Aut}(X)\) is non-reductive (see Corollary 1.2).
Based on the main classification result, one can also derive that if \(X\) is a Du Val del Pezzo surface with \(K_{X}^{2} \geq 3\) and if \(\tau(X)\) denotes its Fano-Weil index, then \(\tau(X) > 1\) implies that the automorphism group \(\mathrm{Aut}(X)\) is infinite.
Reviewer: Piotr Pokora (Kraków)On the eigenpoints of cubic surfaces.https://www.zbmath.org/1456.140432021-04-16T16:22:00+00:00"Celik, Turku Ozlum"https://www.zbmath.org/authors/?q=ai:celik.turku-ozlum"Galuppi, Francesco"https://www.zbmath.org/authors/?q=ai:galuppi.francesco"Kulkarni, Avinash"https://www.zbmath.org/authors/?q=ai:kulkarni.avinash"Sorea, Miruna-Ştefana"https://www.zbmath.org/authors/?q=ai:sorea.miruna-stefanaThe aim of the paper is to study the eigenscheme of order three partially symmetric and symmetric tensors. They also show that a subvariety of the Grassmannian \(Gr(3,\mathbb{P}^{14})\) parametrizes the eigenscheme of \(4 \times 4 \times 4\) symmetric tensors.
The spectral theory of tensors is a multi-linear generalization of the study of eigenvalues and eigenvectors in the case of matrices. The eigenscheme \(E(\mathcal{T})\) of a tensor \(\mathcal{T}\) can be roughly thought as the set of eigenpoints of the tensor, i.e. eigenvectors of \(\mathbb{C}^{n+1}\) of a particular contraction of the tensor. In the case of partial symmetric or symmetric tensors of order \(3\) a contraction may be the following. A partial symmetric tensor \(\mathcal{T} \in \operatorname{Sym}^2 \mathbb{C}^{n+1} \otimes \mathbb{C}^{n+1}\) can be seen as an \((n+1)\)-tuple of quadratic forms \((q_0,\dots,q_n)\) in the variables \(x_i\). Analogously, given a a symmetric tensor \(f \in \operatorname{Sym}^3 \mathbb{C}^{n+1}\), i.e. a homogeneous cubic polynomial, one can associate to it an \((n+1)\)-tuple of quadratic forms given by its derivatives \(\frac{\partial f}{\partial x_i}\). The authors investigates the eigenscheme and some its particular subschemes of the aforementioned tensors with those contractions.
At first they recall some basic notions regarding the theory. In particular they introduce the irregular eigenscheme \(\operatorname{Irr}(\mathcal{T})\) and the regular eigenscheme \(\operatorname{Reg}(\mathcal{T})\). The first can be thought as the subscheme of \(E(\mathcal{T})\) given by points with zero eigenvalue, while the second is the residue of \(E(\mathcal{T})\) with respect to \(\operatorname{Irr}(\mathcal{T})\). After that they focus on the case of order \(3\) symmetric tensors providing bounds on the dimensions and geometric properties of the irregular and regular eigenschemes. Numerous examples of symmetric tensors satisfying all the described properties are provided. As they observe, if the regular eigenscheme of a cubic polynomial is \(0\) dimensional, then it consists of at most \(2^{n+1}-1\) points. Therefore they investigate in the ternary and quaternary case whether there exists a cubic polynomial with a prescribed number of regular eigenpoints. Eventually they show that a open subvariety of a linear subspace of the Grassmannian \(Gr(3,\mathbb{P}^{14})\) parametrizes the eigenschemes of order \(3\) quaternary symmetric tensors.
Reviewer: Reynaldo Staffolani (Trento)Algebraic cycles on moduli space of polarized hyperkähler manifolds.https://www.zbmath.org/1456.140122021-04-16T16:22:00+00:00"Li, Zhiyuan"https://www.zbmath.org/authors/?q=ai:li.zhiyuanSummary: We discuss the recent progress on studying algebraic cycle classes on the moduli space of polarized \(K3\) surfaces and more generally, polarized hyperkähler manifolds. This includes the construction of the tautological ring on these moduli spaces using Noether-Lefschetz theory and Kappa classes defined by Marian-Oprea-Pandaripande. These concepts have been recently applied to study the Chow groups and cohomology groups of these moduli spaces via various methods. In particular, we will discuss some fundamental questions and conjectures, such as the Noether-Lefschetz conjecture, tautological conjecture and generalized Franchetta conjecture. This paper is contributed to the first annual meeting of ICCM.
For the entire collection see [Zbl 1454.00056].On rigid germs of finite morphisms of smooth surfaces.https://www.zbmath.org/1456.140412021-04-16T16:22:00+00:00"Kulikov, Vik. S."https://www.zbmath.org/authors/?q=ai:kulikov.viktor-sA germ of a finite morphism of smooth surfaces is rigid if any germ of cover, which is deformation equivalent to it, is equivalent to it. The author shows that a germ of a finite morphism of smooth surfaces is rigid if and only if the germ of its branch curve has only singularities of type \(A_n\), \(n\geq 1\), \(D_n\), \(n\geq 4\), \(E_6\), \(E_7\) or \(E_8\).
A rational function defined over the algebraic closure of the field of rational numbers is called a Belyi function if it has not more than three critical values.
The author establishes a correspondence between the set of rigid germs of finite morphisms and the set of Belyi rational functions.
Reviewer: Vladimir P. Kostov (Nice)On \(\alpha '\)-effects from \(D\)-branes in \(4d\) \( \mathcal{N} = 1\).https://www.zbmath.org/1456.813402021-04-16T16:22:00+00:00"Weissenbacher, Matthias"https://www.zbmath.org/authors/?q=ai:weissenbacher.matthiasSummary: In this work we study type IIB Calabi-Yau orientifold compactifications in the presence of space-time filling D7-branes and O7-planes. In particular, we conclude that \(\alpha'^2g_s\)-corrections to their DBI actions lead to a modification of the four-dimensional \(\mathcal{N} = 1\) Kähler potential and coordinates. We focus on the one-modulus case of the geometric background i.e. \(h^{1,1} = 1\) where we find that the \(\alpha'^2g_s \)-correction is of topological nature. It depends on the first Chern form of the four-cycle of the Calabi-Yau orientifold which is wrapped by the D7-branes and O7-plane. This is in agreement with our previous F-theory analysis and provides further evidence for a potential breaking of the no-scale structure at order \(\alpha'^2g_s\). Corrected background solutions for the dilaton, the warp-factor as well as the internal space metric are derived. Additionally, we briefly discuss \(\alpha '\)-corrections from other \(Dp\)-branes.Nef cone and Seshadri constants on products of projective bundles over curves.https://www.zbmath.org/1456.140532021-04-16T16:22:00+00:00"Karmakar, Rupam"https://www.zbmath.org/authors/?q=ai:karmakar.rupam"Misra, Snehajit"https://www.zbmath.org/authors/?q=ai:misra.snehajitIn the paper under review the authors study the geometry of the nef cone and the Seshadri constants on products of projective bundles over curves. Let \(E_{1}\) and \(E_{2}\) be two vector bundles over a smooth curve \(C\) of rank \(r_{1}\) and \(r_{2}\), respectively, and degree \(d_{1}\) and \(d_{2}\). Let \(\mathbb{P}(E_{i}) =\mathrm{Proj}(\bigoplus_{d\geq 0} \mathrm{Sym}^{d} (E_{i}))\). Consider the fiber product \(X = \mathbb{P}(E_{1}) \times_{C} \mathbb{P}(E_{2})\) over \(C\) and define \(p_{i} : X \rightarrow \mathbb{P}(E_{i})\) for \(i \in \{1,2\}\). Denote by \(\eta_{i} = [\mathcal{O}_{\mathbb{P}(E_{i})}(1)] \in N^{1}(\mathbb{P}(E_{i}))\). The first result of the paper provides us a full description of the nef cone of \(X\).
Theorem A. Let \(E_{1}, E_{2}\) be two vector bundles on a smooth complex curve \(C\), then
\[\mathrm{Net}(X) = \{ a\tau_{1} + b\tau_{2} + cF \, : \, a,b,c \in \mathbb{R}_{\geq 0}\},\]
where \(\tau_{1} =p_{2}^{*} \eta_{1} - \mu_{11}F\) and \(\tau_{2} = p_{1}^{*} \eta_{2} - \mu_{21}F\) and \(F\) is nef, where \(\mu_{11}, \mu_{21}\) are the smallest slopes of any torsion-free quotients of \(E_{1}\) and \(E_{2}\), respectively (in the sense of the Harder-Narasimhan filtration).
Let us recall that if \(X\) is a smooth complex projective variety and \(L\) a nef line bundle on \(X\), then the Seshadri constant of \(L\) at a point \(x \in X\) is defined as
\[\varepsilon(X,L;x) = \mathrm{inf}_{C \subset X} \bigg\{ \frac{L \cdot C}{\mathrm{mult}_{x} (C)}\bigg\},\]
where the infimum is taken over all irreducible curves in \(X\) passing through \(X\) having multiplicity \(\mathrm{mult}_{x}(C)\) at \(X\).
Theorem B. Let \(E_{1}, E_{2}\) be two vector bundles on a smooth curve \(C\) with \(\mu_{11}, \mu_{21}\) being the smallest slopes of any torsion-free quotient of \(E_{1}\) and \(E_{2}\), respectively. Let \(L\) be an ample line bundle on \(X\) which is numerically equivalent to \(a \tau_{1} + b\tau_{2} + cF \in N^{1}(X)\). Then the Seshadri constants of \(L\) satisfy, for any \(x \in X\), the following inequality
\[\varepsilon(X,L;x) \geq\min\{a,b,c\}.\]
Moreover, if \(a = \min \{a,b,c\}\), then \(\varepsilon(X,L;x) = a\) for any \(x \in X\), or if \(b = \min\{a,b,c\}\), then \(\varepsilon(X,L;x) = b\) for any \(x \in X\).
Reviewer: Piotr Pokora (Kraków)On coefficients of Poincaré series and single-valued periods of modular forms.https://www.zbmath.org/1456.110672021-04-16T16:22:00+00:00"Fonseca, Tiago J."https://www.zbmath.org/authors/?q=ai:fonseca.tiago-jSummary: We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level \(\varGamma_0(N)\) and integral weight \(k\ge 2\) coincides with the field generated by the single-valued periods of a certain motive attached to \(\varGamma_0(N)\). This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of
\textit{F. Brown} [Res. Math. Sci. 5, No. 3, Paper No. 34, 36 p. (2018; Zbl 1440.11071)] and Acres-Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann-Ono's construction of harmonic lifts of Poincaré series [\textit{K. Bringmann} and \textit{K. Ono}, Proc. Natl. Acad. Sci. USA 104, No. 10, 3725--3731 (2006; Zbl 1191.11013)].Intersection cohomology of pure sheaf spaces using Kirwan's desingularization.https://www.zbmath.org/1456.140152021-04-16T16:22:00+00:00"Chung, Kiryong"https://www.zbmath.org/authors/?q=ai:chung.kiryong"Yoon, Youngho"https://www.zbmath.org/authors/?q=ai:yoon.younghoLet \(\mathbf{M}_n\) be the space parametrizing semi-stable sheaves \(F\) on \(\mathbb {P}^n\) with a linear resolution \[0\to\mathcal {O}_{\mathbb {P}^n}(-1)^2 \to \mathcal {O}_{\mathbb {P}^n}^2\to F\to 0.\] \(\mathbb {M}_n\) is an integral normal variety, \(\dim \mathbb {M}_n =4n-3\), which is the Simpsons compactification of twisted sheaves \(\mathcal{I}_{L,Q}(1)\), where \(Q\subset \mathbb {P}^n\) is a rank \(4\) hyperquadric and \(L\subset Q\) is a linear subspace of dimension \(n-2\). The authors computes the intersection Poincaré polynomial of \(\mathbf{M}_n\) using Kirwan's desingularization method and the relation between \(\mathbf{M}_n\), the GIT quotient of the Kroneker quiver (Kontsevich's map space \(\mathbf{K}_n\)). Then they compute the intersection Poincaré polynomial of the moduli space of pure one-dimensional sheaves on the smooth surfaces \(\mathbb {P}^2\), \(\mathbb{F}_0\) and \(\mathbb {F}_1\).
Reviewer: Edoardo Ballico (Povo)Differential forms and quadrics of the canonical image.https://www.zbmath.org/1456.140132021-04-16T16:22:00+00:00"Rizzi, Luca"https://www.zbmath.org/authors/?q=ai:rizzi.luca"Zucconi, Francesco"https://www.zbmath.org/authors/?q=ai:zucconi.francescoSummary: We extend the theory of \textit{G. P. Pirola} and \textit{F. Zucconi} [J. Algebr. Geom. 12, No. 3, 535--572 (2003; Zbl 1083.14515)]. We introduce the new notion of adjoint quadric for canonical images of irregular varieties. Using this new notion, we obtain the infinitesimal Torelli theorem for varieties whose canonical image is a complete intersection of hypersurfaces of degree \(>2\) and for Schoen surfaces. Finally, we show that a family with fiberwise liftable holomorphic forms such that the fibers have Albanese morphism of degree 1 is birationally trivial if there exist no adjoint quadrics.Distributions of extremal black holes in Calabi-Yau compactifications.https://www.zbmath.org/1456.830422021-04-16T16:22:00+00:00"Hulsey, George"https://www.zbmath.org/authors/?q=ai:hulsey.george"Kachru, Shamit"https://www.zbmath.org/authors/?q=ai:kachru.shamit"Yang, Sungyeon"https://www.zbmath.org/authors/?q=ai:yang.sungyeon"Zimet, Max"https://www.zbmath.org/authors/?q=ai:zimet.maxSummary: We study non-supersymmetric extremal black hole excitations of 4d \(\mathcal{N} = 2\) supersymmetric string vacua arising from compactification on Calabi-Yau threefolds. The values of the (vector multiplet) moduli at the black hole horizon are governed by the attractor mechanism. This raises natural questions, such as ``what is the distribution of attractor points on moduli space?'' and ``how many attractor black holes are there with horizon area up to a certain size?'' We employ tools developed by \textit{F. Denef} and \textit{M. R. Douglas} [``Distributions of flux vacua'', J. High Energy Phys. 2004, No. 5, Paper No. 072, 46 p. (2004; \url{doi:10.1088/1126-6708/2004/05/072})] to answer these questions.Stability and Fourier-Mukai transform on elliptic threefolds.https://www.zbmath.org/1456.140222021-04-16T16:22:00+00:00"Lo, Jason"https://www.zbmath.org/authors/?q=ai:lo.jason|lo.jason.1Summary: This article is based on an invited talk of the same title, given at the second annual meeting of the International Consortium of Chinese Mathematicians (ICCM) in Taipei, December 2018. After a quick survey of the concepts of slope stability, Bridgeland stability and polynomial stability, the notion of a scale is introduced as an intermediary between a heart and a slicing. Drawing from the case of an elliptic curve as a motivation, we expand the use of scales to elliptic threefolds. This forms a key step in the proof of a preservation of stability result under a relative Fourier-Mukai transform on elliptic threefolds.
For the entire collection see [Zbl 1454.00057].A construction of infinitely many solutions to the Strominger system.https://www.zbmath.org/1456.813342021-04-16T16:22:00+00:00"Fei, Teng"https://www.zbmath.org/authors/?q=ai:fei.teng.1|fei.teng"Huang, Zhijie"https://www.zbmath.org/authors/?q=ai:huang.zhijie"Picard, Sebastien"https://www.zbmath.org/authors/?q=ai:picard.sebastienFrom the introduction:: As for compact Kähler Calabi-Yau manifolds (treated as solutions to
the Strominger system), it is widely speculated that in each dimension
there are only finitely many deformation types and hence finitely many
sets of Hodge numbers. Moreover, there are no explicit expressions for
Calabi-Yau metrics except for the flat case.\par\vspace{1mm}
In this paper, we demonstrate that the non-Kähler world of solu-
tions to the Strominger system is considerably different. More pre-
cisely, we construct explicit smooth solutions to the Strominger system
on compact non-Kähler Calabi-Yau 3-folds with infinitely many topo-
logical types and sets of Hodge numbersSeshadri constants of the anticanonical divisors of Fano manifolds with large index.https://www.zbmath.org/1456.140112021-04-16T16:22:00+00:00"Liu, Jie"https://www.zbmath.org/authors/?q=ai:liu.jie.7|liu.jie|liu.jie.4|liu.jie.6|liu.jie.5|liu.jie.2|liu.jie.3|liu.jie.1Let \(X\) be a normal complex projective variety, \(L\) a nef line bundle over \(X\), and \(x \in X\).
The Seshadri constant \(\epsilon(X,L;x)\) is the infimum over all irreducible cuves \(C \subset X\)
of the quotients of the \(L\)-degree of \(C\) by the multiplicity of \(C\) at \(x\). It measures the
local positivity of \(L\) at \(x\) and, as a function over \(X\), is a lower-continuous function (in the
topology whose closed subsets are countable union of Zariski closed sets). The maximum value,
attained at a very general \(x \in X\), is denoted as \(\epsilon(X,L;1)\), and is upper
bounded by \(\sqrt[n]{L^n}\). Several lower bounds are known when \(L\) ample
(see the Introduction of the paper under
review and references therein), and it is conjectured to be lower bounded by one when \(X\) smooth
and \(L\) ample (see Conj. 1.2). In the particular case of \(X\) a Fano manifold, and \(L\) the
anticanonical bunde, one can consider the question of classifying Fano manifolds for which
\(\epsilon(X,-K_X,1) \leq 1\). The list is known for Del Pezzo surfaces (see Theorem 1.4) and they are
exactly the ones for which the linear system \(|-K_X|\) is not base point free. In Theorem 1.5 the
author extends some previously known results to show that when the index \(r_X\) of the
Fano variety \(X\) is greater than or equal to \(\dim(X)-3\) then \(\epsilon(X,-K_X;1) \geq r_X\) as predicted.
In Theorem 1.6, it is shown that when the index is greater than or equal to the maximum of \(2\) and
\(\dim(X)-2\) then \(\epsilon(X,-K_X;1) =r_X\) and also equal to the minimal anticanonical degree of
a covering family of rational curves. Finally, in Theorem 1.7, a explicit computation of \(\epsilon(X,-K_X;1)\)
for smooth Fano threefolds with Picard number greater than or equal to two is provided.
This, together with previously known results lead to the corollary (see Cor. 1.8) that for \(X\) a smooth Fano threefolds very general in
its deformation family, \(\epsilon(X,-K_X;1) \leq 1\) is equivalent to the fact that \(|-K_X|\)
is not base point free. Examples in dimension \(4\) show (see Ex. 1.10) that the same result does
not hold in higher dimension but the question on the non-emptyness of
the base locus of \(|-K_X|\) for Fano manifolds such that \(\epsilon(X,-K_X;1) \leq 1\)
is posed.
Reviewer: Roberto Muñoz (Madrid)Quiver Yangian from crystal melting.https://www.zbmath.org/1456.812162021-04-16T16:22:00+00:00"Li, Wei"https://www.zbmath.org/authors/?q=ai:li.wei.7|li.wei-wayne|li.wei.10|li.wei.11|li.wei.9|li.wei.4|li.wei.8|li.wei|li.wei.3|li.wei.2"Yamazaki, Masahito"https://www.zbmath.org/authors/?q=ai:yamazaki.masahitoSummary: We find a new infinite class of infinite-dimensional algebras acting on BPS states for non-compact toric Calabi-Yau threefolds. In Type IIA superstring compactification on a toric Calabi-Yau threefold, the D-branes wrapping holomorphic cycles represent the BPS states, and the fixed points of the moduli spaces of BPS states are described by statistical configurations of crystal melting. Our algebras are ``bootstrapped'' from the molten crystal configurations, hence they act on the BPS states. We discuss the truncation of the algebra and its relation with D4-branes. We illustrate our results in many examples, with and without compact 4-cycles.Ordinary K3 surfaces over a finite field.https://www.zbmath.org/1456.140442021-04-16T16:22:00+00:00"Taelman, Lenny"https://www.zbmath.org/authors/?q=ai:taelman.lennyThe author, building on [\textit{N. O. Nygaard}, Progr. Math. No 35, p. 267--276 (1983; Zbl 0574.14031)] and [\textit{J.-D. Yu} Pure Appl. Math. Q. 8, No. 3, 805--824 (2012; Zbl 1252.14008)], studies the category of ordinary \(K3\) surfaces over a finite field.
A \(K3\) surface \(X\) over a finite field \(k\) of characteristic \(p>0\) is called \emph{ordinary} if \(| X(k) | \not\equiv 1 \bmod p\). Nygaard and Yu already exhibited the existence of a fully faithful functor between the groupoids of
\begin{itemize}
\item[1.] the ordinary \(K3\) surfaces over a finite field \(\mathbb{F}_q\) and
\item[2.] the triples \((M, F, \mathcal{K})\), consisting of
\begin{itemize}
\item[(a)] an integral lattice \(M\),
\item[(b)] an endomorphism \(F\) of \(M\), and
\item[(c)] a convex subset of \(\mathcal{K} \subset \mathbb{R}\otimes M\)
\end{itemize}
satisfying certain conditions.
\end{itemize}
The main result of the paper under review consists in studying the image of this functor and, in particular, in showing that if every \(K3\) surface over the fraction field of the ring of Witt vectors \(W(\mathbb{F}_q)\) satisfies a strong form of ``potential semi-stable reduction'', then the functor is essentially surjective. Finally, using this property, the author describes three sub-grupoids on which the functor restricts to an equivilance.
Reviewer: Dino Festi (Mainz)Categorical localization for the coherent-constructible correspondence.https://www.zbmath.org/1456.140472021-04-16T16:22:00+00:00"Ike, Yuichi"https://www.zbmath.org/authors/?q=ai:ike.yuichi"Kuwagaki, Tatsuki"https://www.zbmath.org/authors/?q=ai:kuwagaki.tatsukiKontsevich's homological mirror symmetry(HMS) conjecture states that two categories associated to
a mirror pair are equivalent. For a Calabi-Yau(CY) variety, a mirror is also Calabi-Yau and the conjecture is a
quasi-equivalence between the dg category of coherent sheaves over one and the derived Fukaya category of
the other. For non-CY's, mirrors do not need to be varieties. For a Fano toric variety, its mirror is a
Landau-Ginzburg (LG) model, which is a holomorphic function on \((\mathbb C^\times)^n\) which can be read from
the defining fan of the toric variety which is in fact the specialization of Lagrangian potential
function of the toric \(A\)-model that is the generating function of open Gromov-Witten invariants of
a toric fiber [\textit{C.-H. Cho} and \textit{Y.-G. Oh}, Asian J. Math. 10, No. 4, 773--814 (2006; Zbl 1130.53055); \textit{K. Fukaya} et al., Duke Math. J. 151, No. 1, 23--175 (2010; Zbl 1190.53078)].
For a smooth Fano, it has been proven for many special cases that the dg category of coherent sheaves
\(\mathbf{coh}\, X_\Sigma\) over the toric variety \(X_\Sigma\) associated to the fan \(\Sigma\) is quasi-equivalent to
the Fukaya-Seidel category \(\mathfrak{Fuk}(W_\Sigma)\) of the associated Laurent polynomial \(W_\Sigma\).
When a variety is not complete, \(\mathbf{coh}\, X_\Sigma\) is of infinite dimensional nature and its Fukaya-type
category also should have an infinite-dimensional nature. Such a construction is known to be
(partially) wrapped Fukaya categories. In this regard, the main theme of the present paper in review is
to establish a quasi-isomorphism \(\mathbf{coh}(X\setminus D)\cong \mathbf{coh} X/\mathbf{coh}_D X\) in some special cases
in the microlocal world: Here
\(X\setminus D\) is the complement of a divisor \(D\) and \(\mathbf{coh} X/\mathbf{coh}_D X\) is the dg category of
sheaves supported in \(D\) by relating the isomorphism to a similar isomorphism
\[
W_{\mathbf{s}\setminus\mathbf{r}}(M) \cong W_{\mathbf{s}}(M)/\mathfrak B_{\mathbf{r}}
\]
of \textit{Z. Sylvan} [J. Topol. 12, No. 2, 372--441 (2019; Zbl 1430.53097)] in the Fukaya-Seidel side: Here \(\mathbf{s}\) is a collection
of symplectic stops and \(\mathbf{r} \subset\mathbf{s}\) is a sub-collection thereof, and
\(\mathfrak B_{\mathbf{r}}\) is the full subcategory spanned by Lagrangians near the sub-stops \(\mathbf{r}\).
The paper extends a version of coherent-constructible correspondence [\textit{B. Fang} et al., Invent. Math. 186, No. 1, 79--114 (2011; Zbl 1250.14011); \textit{K. Bongartz} et al., Adv. Math. 226, No. 2, 1875--1910 (2011; Zbl 1223.16004)] to the dg category of
\emph{quasi-coherent shaves} over \(X_\Sigma\) in dimension 2.
Reviewer: Yong-Geun Oh (Pohang)Birational superrigidity and \(K\)-stability of Fano complete intersections of index \(1\).https://www.zbmath.org/1456.140502021-04-16T16:22:00+00:00"Zhuang, Ziquan"https://www.zbmath.org/authors/?q=ai:zhuang.ziquanA Fano variety \(X\) is said to be birationally superrigid if it has terminal singularities,
it is \({\mathbb Q}\)-factorial of Picard number 1, and every birational map \(X\)
to a Mori fiber space is an isomorphism. On the other hand, \(X\) is \(K\)-stable
with respect to its anticanonical bundle if, essentially, it admits a Kähler-Einstein
metric, and \(K\)-stability is encoded in the positivity of the invariants \(\beta(F)\)
for \(F\) any dreamy prime divisor \(F\) over \(X\) (see 2.2 for details). In the paper under
review the author shows (see Thm. 1.2 and 1.3) that for a \(n\)-dimensional smooth Fano complete
intersection \(X \subset {\mathbb P}^{n+r}\) of index one, if \(n \geq 10r\) then \(X\) is birationally
superrigid and \(K\)-stable. Moreover, the smooth complete intersection of a quadric and a cubic
in \({\mathbb P}^5\) is also \(K\)-stable. For a Fano manifold (see Def. A.1 in the appendix of
the paper under review) \(X\) is said to be conditionally birationally superrigid if
every birational map from \(X\) to a Mori fiber space whose undefined locus has
codimension at least \(1\) plus the index of \(X\) is an isomorphism.
In the Appendix, the authors show that Fano complete intersections of higher index in large dimension (see Cor. A.3
for details) are conditionally birationally superrigid.
Reviewer: Roberto Muñoz (Madrid)Correspondence scrolls.https://www.zbmath.org/1456.140482021-04-16T16:22:00+00:00"Eisenbud, David"https://www.zbmath.org/authors/?q=ai:eisenbud.david"Sammartano, Alessio"https://www.zbmath.org/authors/?q=ai:sammartano.alessioThis paper introduces schemes called ``correspondence scrolls'' and gives a general study of them. For a closed subscheme \(Z\) in \(\Pi _{i=1}^n {\mathbb A}^{a_i+1}\), defined by a multigraded ideal \(I\subset A:={\Bbbk}[x_{i,j} : 1\le i \le n, 0 \le j \le a_i]\), and for \(\mathbf{b}=(b_1, \ldots , b_n) \in \mathbb{N}_+^n\), consider the homomorphism \({\Bbbk}[z_{i,\alpha }] \rightarrow A/I\) which sends a variable \(z_{i, \alpha }\) to the monomial \(x_i^\alpha\), of degree \(b_i\). Here \(x_i^ {\alpha }\) denotes \(x_{i,0}^{\alpha _0} \cdots x_{i, a_i}^{\alpha _{a_i}}\). The kernel of the above map defines a closed projective subscheme \(C(Z, {\mathbf b}) \subset { \mathbb P}^N\) (\(N= \sum \binom{a_i+b_i}{a_i}-1)\), called \textit{correspondence scroll}. This definition includes classical correspondences as well as interesting non-classical ones: rational normal scrolls, double structures which are degenerate \(K3\) surfaces, degenerate Calabi-Yau threefolds, etc. Many invariants or properties of correspondence scrolls are studied: dimension, degree, nonsingularity, Cohen-Macaulay and Gorenstein property and others. The paper is very well written and invites to further research.
Reviewer: Nicolae Manolache (Bucureşti)On TCS \(G_2\) manifolds and 4D emergent strings.https://www.zbmath.org/1456.831092021-04-16T16:22:00+00:00"Xu, Fengjun"https://www.zbmath.org/authors/?q=ai:xu.fengjunSummary: In this note, we study the Swampland Distance Conjecture in TCS \(G_2\) manifold compactifications of M-theory. In particular, we are interested in testing a refined version --- the Emergent String Conjecture, in settings with 4d \(N = 1\) supersymmetry. We find that a weakly coupled, tensionless fundamental heterotic string does emerge at the infinite distance limit characterized by shrinking the \(K3\)-fiber in a TCS \(G_2\) manifold. Such a fundamental tensionless string leads to the parametrically leading infinite tower of asymptotically massless states, which is in line with the Emergent String Conjecture. The tensionless string, however, receives quantum corrections. We check that these quantum corrections do modify the volume of the shrinking \(K3\)-fiber via string duality and hence make the string regain a non-vanishing tension at the quantum level, leading to a decompactification. Geometrically, the quantum corrections modify the metric of the classical moduli space and are expected to obstruct the infinite distance limit. We also comment on another possible type of infinite distance limit in TCS \(G_2\) compactifications, which might lead to a weakly coupled fundamental type II string theory.Hyperkähler cones and instantons on quaternionic Kähler manifolds.https://www.zbmath.org/1456.530402021-04-16T16:22:00+00:00"Devchand, Chandrashekar"https://www.zbmath.org/authors/?q=ai:devchand.chandrashekar"Pontecorvo, Massimiliano"https://www.zbmath.org/authors/?q=ai:pontecorvo.massimiliano"Spiro, Andrea"https://www.zbmath.org/authors/?q=ai:spiro.andrea-fIn this paper a new method to construct Yang-Mills instantons over quaternionic pseudo-Riemannian Kähler manifolds is introduced by extending a technique motivated by supersymmetry for constructing Yang-Mills instantons over pseudo-Riemannian hyper-Kähler manifods by the same authors.
By a \textit{quaternionic pseudo-Riemannian Kähler manifold} of signature \((p,q)\) one means a real \(4n\)-dimensional pseudo-Riemannian manifold \((M,g)\) whose holonomy group is isomorphic to \(\mathrm{Sp}(1)\mathrm{Sp}(p,q)\) satisfying \(p+q=n\). Such manifolds are automatically Einstein hence solving the Yang-Mills self-duality equations over them is interesting from both a mathematical and a physical viewpoint. The authors' method in the spirit of the classical Atiyah-Ward correspondence rests on a bijection between gauge equivalence classes of Yang-Mills instantons with arbitrary compact structure group \(G\) over \((M,g)\) and certain holomorphic objects over a twistor space-like complex manifold \(H(S(M))\) associated to \((M,g)\). This twistor space is constructed in two steps. First, given \((M,g)\) one takes the so-called \textit{Swann bundle} over \(M\), i.e., a certain \({\mathbb H}^*/{\mathbb Z}_2\)-bundle \(\pi :S(M)\rightarrow M\). It has the structure of a hyper-Kähler cone over \(M\). Secondly, one considers its \textit{harmonic space} \(H(S(M))\) which is a topologically trivial \(\mathrm{SL}(2;{\mathbb C})\)-bundle over \(S(M)\) carrying the unique non-product complex structure which makes \(H(S(M))\) a holomorphic bundle over the classical twistor space \(Z(S(M))=S(M)\times{\mathbb C}P^1\) of the Swann bundle; the fibers of this vector bundle are isomorphic to the Borel subgroup \(B\subset\mathrm{SL}(2;{\mathbb C})\) since \(\mathrm{SL}(2;{\mathbb C})/B\cong{\mathbb C}P^1\). The key observation is that there is a one-to-one correspondence between (local) \(G\)-instantons over \((M,g)\) and certain (local) holomorphic maps, called (supersymmetric) \textit{prepotentials}, from \(H(S(M))\) into the complexified Lie algebra of \(G\). Combining this method with results of Narasimhan and Ramanan the construction settles down to a set of data on certain local maps \(M\supset U\rightarrow \mathrm{Mat}_{k\times m} ({\mathbb C})\) and in this form the construction resembles the classical ADHM construction.
Reviewer: Gabor Etesi (Budapest)From cracked polytopes to Fano threefolds.https://www.zbmath.org/1456.140492021-04-16T16:22:00+00:00"Prince, Thomas"https://www.zbmath.org/authors/?q=ai:prince.thomas-a|prince.thomas|prince.thomas-mFix a complete (generalized) fan \(\Sigma\). In his previous paper [Manuscr. Math. 163, No. 1--2, 165--183 (2020; Zbl 07233347)], the author introduced the notion of cracked polytope along \(\Sigma\) as a polytope whose intersection with each maximal cone of \(\Sigma\) is unimodular. In the present paper, starting from a cracked polytope and using Laurent inversion, the author constructs Fano threefolds with very ample anticanonical bundle and Picard rank greater than one. He also gives constructions, from cracked polytopes, of rank one Fano threefolds. Moreover, he investigates the problem of classifying polytopes cracked along a given fan in three dimensions, and he classifies the unimodular polytopes which can occur as pieces of a cracked polytope.
Reviewer: Carla Novelli (Padova)Hypersurfaces with linear type singular loci.https://www.zbmath.org/1456.130082021-04-16T16:22:00+00:00"Farrahy, Amir Behzad"https://www.zbmath.org/authors/?q=ai:farrahy.amir-behzad"Nasrollah Nejad, Abbas"https://www.zbmath.org/authors/?q=ai:nasrollah-nejad.abbasThe authors consider an affine hypersurface \(X\subset\mathbb A^n\) defined by a polynomial \(f\) in the ring \(R=k[x_1,\dots,x_n]\) over an algebraically closed field \(k\) of characteristic \(0\). The singular locus of \(X\) is defined by the vanishing of the Jacobian ideal \(I(f)\), which is generated by \(f\) and the \(n\) derivatives of \(f\). The ideal \(J(f)\) generated by the derivatives alone is the gradient ideal of \(f\). The authors compare the schemes defined by \(I(f)\) and by \(J(f)\) around a singular point \(p\) of \(X\), assuming that \(X\) has only isolated singularities (so that, in particular, it is reduced). The Milnor number of \(X\) at \(p\) is the dimension of the localization \((R/J(f))_p\), while the Tjurina number of \(X\) at \(p\) is defined as the dimension of the localization \((R/I(f))_p\). \(X\) is called `locally Eulerian' if the Milnor number is equal to the Tjurina number at each singular point. The authors prove that \(X\) is locally Eulerian if and only if the symmetric and the Rees algebras of \(I(f)\) in \(R\) are isomorphic. In turn, the condition is equivalent to say that the Tjurina algebra \(R/I(f)\) is artinian Gorenstein. Then, the authors extend their study to the case of reduced projective plane curves \(C\), defining \(C\) to be `of gradient type' if the restriction of \(C\) to each affine chart associated to the singular points is locally Eulerian. The authors prove that curves \(C\) with only simple singularities are of gradient type. Furthermore, they prove that (reduced) quartics are of gradient type, while there are examples of quintics and sextics which are not of gradient type.
Reviewer: Luca Chiantini (Siena)A hypergeometric version of the modularity of rigid Calabi-Yau manifolds.https://www.zbmath.org/1456.110732021-04-16T16:22:00+00:00"Zudilin, Wadim"https://www.zbmath.org/authors/?q=ai:zudilin.wadimThis paper considers the fourteen one-parameter families of Calabi-Yau
threefolds whose periods are expressed in terms of hypergeometric functions.
For these fourteen families, periods are solutions of hypergeometric equations
with parameter \((r, 1-r, t, 1-t)\), where
\begin{multline*}
(r,t)=\Big(\frac{1}{2},\frac{1}{2}\Big),\Big(\frac{1}{2},\frac{1}{3}\Big),\Big(\frac{1}{2},\frac{1}{4}\Big),
\Big(\frac{1}{2},\frac{1}{6}\Big),\Big(\frac{1}{3}\Big),\Big(\frac{1}{3},\frac{1}{4}\Big),\Big(\frac{1}{3},\frac{1}{6}\Big),\\
\Big(\frac{1}{4},\frac{1}{4}\Big),\Big(\frac{1}{4},\frac{1}{6}\Big),\Big(\frac{1}{6},\frac{1}{6}\Big),\Big(\frac{1}{5},\frac{2}{5}\Big),
\Big(\frac{1}{8},\frac{3}{8}\Big),\Big(\frac{1}{10},\frac{3}{10}\Big),\Big(\frac{1}{12},\frac{5}{12}\Big).
\end{multline*}
At a conifold point, any of these Calabi-Yau threefolds becomes rigid,
and the \(p\)-th coefficient \(a(p)\) of the corresponding modular form of weight \(4\)
can be recovered from the truncated partial sums of the corresponding
hypergeometric series modulo a higher power of \(p\), where \(p\) is any good prime \(>5\).
This paper discusses relationships between the critical values of the \(L\)-series of the modular form
and the values of a related basis of solutions to the hypergeometric differential equation.
It is numerically observed that the critical \(L\)-values are \(\mathbb{Q}\)-proportional to the
hypergeometric values \(F_1(1), F_2(1), F_3(1)\), where \(F_j(z)\) are solutions of the hypergeometric
equation for the hypergeometric function \(F_0(z)=_4F_3(z)\) with parameters \((r, 1-r, t, 1-t)\).
This confirms the prediction of Golyshev concerning gamma structures [\textit{V. Golyshev} and \textit{A. Mellit}, J. Geom. Phys. 78, 12--18 (2014; Zbl 1284.33001)].
Reviewer: Noriko Yui (Kingston)Singularities of Rees-like algebras.https://www.zbmath.org/1456.130132021-04-16T16:22:00+00:00"Mantero, Paolo"https://www.zbmath.org/authors/?q=ai:mantero.paolo"Miller, Lance Edward"https://www.zbmath.org/authors/?q=ai:miller.lance-edward"McCullough, Jason"https://www.zbmath.org/authors/?q=ai:mccullough.jasonIf \(X\) is a non-degenerate, irreducible, embedded projective variety over an algebraically closed field \(k\) corresponding to a homogeneous prime ideal \(P \subseteq S = k[x_1,\dots,x_n]\), the Eisenbud-Goto conjecture predicts an upper bound for the regularity of \(X\): \(\hbox{reg}(X) \leq \deg(X) - \hbox{codim}(X) + 1\). This was open for many years, with special cases proved. Thus it was a bit of a shock when Peeva and the second author gave counterexamples to this conjecture, producing irreducible projective varieties with regularity much larger than their degrees. The two main new ideas in their work were so-called Rees-like algebras and step-by-step homogenization. All of the varieties thus produced are singular, and the current paper studies the singularities and their geometry. The authors compute the codimension of the singular locus of a Rees-like algebra over a polynomial ring, and then show that the step-by-step process can decrease the codimension of this singular locus. Thus the authors introduce prime standardization, as an alternative to step-by-step homogenization that preserves the codimension of the singular locus. They then look at the regularity of certain smooth hyperplane sections of Rees-like algebras and show that they all satisfy the Eisenbud-Goto conjecture. They also give a characterization of Rees-like algebras of Cohen-Macaulay ideals, and more generally they characterize when Rees-like algebras are seminormal, weakly normal, and in the case of positive characteristic, F-split.
Reviewer: Juan C. Migliore (Notre Dame)Quotients of the square of a curve by a mixed action, further quotients and Albanese morphisms.https://www.zbmath.org/1456.140452021-04-16T16:22:00+00:00"Pignatelli, Roberto"https://www.zbmath.org/authors/?q=ai:pignatelli.robertoThis paper continues the study of product--quotient surfaces, i.e., surfaces that are the quotient of a product of two algebraic curves by a finite group of automorphisms (cf. [\textit{I. Bauer} and \textit{R. Pignatelli}, Groups Geom. Dyn. 10, No. 1, 319--363 (2016; Zbl 1348.14021); \textit{D. Frapporti} and \textit{R. Pignatelli}, Glasg. Math. J. 57, No. 1, 143--165 (2015; Zbl 1330.14069); \textit{I. Bauer} et al., Am. J. Math. 134, No. 4, 993--1049 (2012; Zbl 1258.14043)].
Here the case of mixed surfaces is looked at. A mixed surface is the minimal resolution S of the singularities of a quotient \((C \times C)/G\) of the square of a curve by a finite group of automorphisms that contains elements not preserving the factors.
The main result is a very precise description of the Albanese morphism of \(S\), when \(S\) is irregular (i.e. when \(q(S)>0\)). It is shown that \(S\) has always maximal Albanese dimension when possible (i.e. when \(q(S)\geq 2\)). Then the result is applied to all the semi-isogenous mixed surfaces with \(p_g=q=2\) constructed by \textit{N. Cancian} and \textit{D. Frapporti} [Math. Nachr. 291, No. 2--3, 264--283 (2018; Zbl 1408.14122)].
The main tool used here is studying further quotients \((C \times C)/G' \) where \(G '\) is a group of automorphisms of \(C\times C\) containing \(G\), and relating the Albanese morphism of \(S\) with the Jacobian \(J(C)\).
Reviewer: Margarida Mendes Lopes (Lisboa)A one parameter family of Calabi-Yau manifolds with attractor points of rank two.https://www.zbmath.org/1456.830892021-04-16T16:22:00+00:00"Candelas, Philip"https://www.zbmath.org/authors/?q=ai:candelas.philip"de la Ossa, Xenia"https://www.zbmath.org/authors/?q=ai:de-la-ossa.xenia-c"Elmi, Mohamed"https://www.zbmath.org/authors/?q=ai:elmi.mohamed"van Straten, Duco"https://www.zbmath.org/authors/?q=ai:van-straten.ducoSummary: In the process of studying the \(\zeta\)-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, first studied by Verrill, for which the quartic numerator of the \(\zeta\)-function factorises into two quadrics remarkably often. Among these factorisations, we find \textit{persistent factorisations}; these are determined by a parameter that satisfies an algebraic equation with coefficients in \(\mathbb{Q}\), so independent of any particular prime. Such factorisations are expected to be modular with each quadratic factor associated to a modular form. If the parameter is defined over \(\mathbb{Q}\) this modularity is assured by the proof of the Serre Conjecture. We identify three values of the parameter that give rise to persistent factorisations, one of which is defined over \(\mathbb{Q}\), and identify, for all three cases, the associated modular groups. We note that these factorisations are due a splitting of Hodge structure and that these special values of the parameter are rank two attractor points in the sense of IIB supergravity. To our knowledge, these points provide the first explicit examples of non-singular, non-rigid rank two attractor points for Calabi-Yau manifolds of full SU(3) holonomy. The values of the periods and their covariant derivatives, at the attractor points, are identified in terms of critical values of the \(L\)-functions of the modular groups. Thus the critical \(L\)-values enter into the calculation of physical quantities such as the area of the black hole in the 4D spacetime. In our search for additional rank two attractor points, we perform a statistical analysis of the numerator of the \(\zeta\)-function and are led to conjecture that the coefficients in this polynomial are distributed according to the statistics of random USp(4) matrices.GLSMs for exotic Grassmannians.https://www.zbmath.org/1456.814342021-04-16T16:22:00+00:00"Gu, Wei"https://www.zbmath.org/authors/?q=ai:gu.wei"Sharpe, Eric"https://www.zbmath.org/authors/?q=ai:sharpe.eric-r"Zou, Hao"https://www.zbmath.org/authors/?q=ai:zou.haoSummary: In this paper we explore nonabelian gauged linear sigma models (GLSMs) for symplectic and orthogonal Grassmannians and flag manifolds, checking e.g. global symmetries, Witten indices, and Calabi-Yau conditions, following up a proposal in the math community. For symplectic Grassmannians, we check that Coulomb branch vacua of the GLSM are consistent with ordinary and equivariant quantum cohomology of the space.Floer cohomology, multiplicity and the log canonical threshold.https://www.zbmath.org/1456.140422021-04-16T16:22:00+00:00"McLean, Mark"https://www.zbmath.org/authors/?q=ai:mclean.markThe notions of multiplicity and log canonical threshold are the fundamental notions of
complex hypersurface \(H = \{f = 0\}\) defined by polynomials \(f\) on \(\mathbb C^{n+1}\). The former one is rather classical
which is defined by
\[
\mu_P(f) = \dim_{\mathbb C} \mathcal O_{\mathbb C^n,P}/(\delta f/\delta x_1, \ldots, \delta f/\delta x_n)
\]
at \(P \in H\). The latter notion of log canonical threshold is relative new which is given by
\[
\text{lct}_P(f) = \min \{(E_j) + 1/ \text{ord}_f(E_j): j \in S\}
\]
at \(P \in H\), where \((E_j)_{j \in S}\) are the \emph{resolution divisors} of a log resolution at \(0 \in H\) of the pair
\((\mathbb C^{n+1},H)\), whose precise current definition is
given by \textit{V. V. Shokurov} in birational geometry [Russ. Acad. Sci., Izv., Math. 40, No. 1, 95--202 (1992; Zbl 0785.14023); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 1, 105--201, Appendix 201--203 (1992)]. Both are algebraic invariants of hypersurface singularities
in complex algebraic geometry. The main result of the paper in review proves that these are indeed symplectic invariants of
the hypersurface in that when \(f, \, g: \mathbb C^{n+1} \to \mathbb C\) are two polynomials with isolated singular points at \(0\) with
embedded contactomorphic links, the multiplicity and the log canonical threshold of \(f\) and \(g\) are equal.
The main technical ingredient used to prove this result is to find formulas for the multiplicity and log canonical threshold
in terms of a sequence of fixed-point Floer cohomology groups in symplectic topology. The author does this by constructing a spectral sequence converging to the fixed-point Floer cohomology of any iterate of the Milnor monodromy map whose
\(E^1\) page is explicitly described in terms of a log resolution of \(f\). This spectral sequence is a generalization
of a forumla by \textit{N. A'Campo} [Comment. Math. Helv. 50, 233--248 (1975; Zbl 0333.14008)]. The author first carries out a rather detailed technical
symplectic massaging, called \(\omega\)-regularization, of a germ of the neighborhood of intersections of
the symplectic crossing divisor \((V_i)_{i\in S}\) which are transversally intersecting codimension 2 symplectic submanifolds.
Then he applies the geometric notions of Liouville domains and open-books to construct a contact open book that is
well-behaved such that the mapping torus of the Milnor monodromy map is isotopic to the mapping torus of a
symplectomorphism arising from the open book. The paper provides much details of basic constructions in symplectic topology
that is expected to be useful for other similar future applications of symplectic machinery to complex algebraic geometry.
Reviewer: Yong-Geun Oh (Pohang)Supercongruences arising from hypergeometric series identities.https://www.zbmath.org/1456.110052021-04-16T16:22:00+00:00"Liu, Ji-Cai"https://www.zbmath.org/authors/?q=ai:liu.jicaiThis paper refines a supercongruence of \textit{T. Kilbourn} [Acta Arith. 123, No. 4, 335--348 (2006; Zbl 1170.11008)] about the identity \[ a(p)=p^3-2p^2-7-N(p) \] studied by \textit{S. Ahlgren} and \textit{K. Ono} [J. Reine Angew. Math. 518, 187--212 (2000; Zbl 0940.33002)], by \textit{B. van Geemen} and \textit{N. O. Nygaard} [J. Number Theory 53, No. 1, 45--87 (1995; Zbl 0838.11047)], and by \textit{H. A. Verrill} [CRM Proc. Lecture Notes 19, 333--340. Providence, RI: Amer. Math. Soc. (1999; Zbl 0942.14022] in connection to the modular Calabi-Yau threefold for odd primes \(p\) \[ x+\frac{1}{x}+y+\frac{1}{y}+z+\frac{1}{z}+w+\frac{1}{w}=0 \] associated with truncated hypergeometric series.
Namely, the author establishes that \[a(p) \equiv p \cdot {{_{4}F_3} \left[ \begin{matrix} \frac{1}{2}, & \frac{1}{2}, & \frac{1}{2}, & \frac{1}{2}\;\\ & 1, & \frac{3}{4}, & \frac{5}{4} \end{matrix} \Big| \; 1 \right]}_ \frac{ p-1}{2} \pmod {p^3}\] for any prime \(p \geq 5 \).
In addition, the paper gives a ``human proof'' of a supercongruence already found by the author [J. Math. Anal. Appl. 471, 613--622 (2019; Zbl 1423.11015)], via the Mathematica package \(Sigma\) supplied by \textit{C. Schneider} [Sémin. Lothar. Comb. 56, B56b, 36 p. (2006; Zbl 1188.05001)], as extension of the \(p\)-adic analogue of a Ramanujan's identity conjectured by \textit{L. van Hamme} [Lect. Notes Pure Appl. Math. 192, 223--236 (1997; Zbl 0895.11051)].
Beyond basic properties of the gamma function (both classical and \(p\)-adic), the Taylor's expansion, and the Wolstenholme's congruence, the theorem-proving recalls some results from \textit{W. N. Bailey} [Generalized hypergeometric series. London: Cambridge University Press (1935; Zbl 0011.02303)], from \textit{L. Long} and \textit{R. Ramakrishna} [Adv. Math. 290, 773--808 (2016; Zbl 1336.33018)], and from \textit{F. J. W. Whipple} [Proc. London Math Soc. 24, 247--263 (1926; JFM 51.0283.03)].
Reviewer: Enzo Bonacci (Latina)Dyonic black hole degeneracies in \(\mathcal{N} = 4\) string theory from Dabholkar-Harvey degeneracies.https://www.zbmath.org/1456.830392021-04-16T16:22:00+00:00"Chowdhury, Abhishek"https://www.zbmath.org/authors/?q=ai:chowdhury.abhishek"Kidambi, Abhiram"https://www.zbmath.org/authors/?q=ai:kidambi.abhiram"Murthy, Sameer"https://www.zbmath.org/authors/?q=ai:murthy.sameer"Reys, Valentin"https://www.zbmath.org/authors/?q=ai:reys.valentin"Wrase, Timm"https://www.zbmath.org/authors/?q=ai:wrase.timmSummary: The degeneracies of single-centered dyonic \(\frac{1}{4}\)-BPS black holes (BH) in Type II string theory on \(K3 \times T^2\) are known to be coefficients of certain mock Jacobi forms arising from the Igusa cusp form \(\Phi_{10}\). In this paper we present an exact analytic formula for these BH degeneracies purely in terms of the degeneracies of the perturbative \(\frac{1}{2}\)-BPS states of the theory. We use the fact that the degeneracies are completely controlled by the polar coefficients of the mock Jacobi forms, using the Hardy-Ramanujan-Rademacher circle method. Here we present a simple formula for these polar coefficients as a quadratic function of the \(\frac{1}{2}\)-BPS degeneracies. We arrive at the formula by using the physical interpretation of polar coefficients as negative discriminant states, and then making use of previous results in the literature to track the decay of such states into pairs of \(\frac{1}{2}\)-BPS states in the moduli space. Although there are an infinite number of such decays, we show that only a finite number of them contribute to the formula. The phenomenon of BH bound state metamorphosis (BSM) plays a crucial role in our analysis. We show that the dyonic BSM orbits with \(U\)-duality invariant \(\Delta < 0\) are in exact correspondence with the solution sets of the Brahmagupta-Pell equation, which implies that they are isomorphic to the group of units in the order \(\mathbb{Z} [ \sqrt{\left|\Delta \right|} ]\) in the real quadratic field \(\mathbb{Q} ( \sqrt{\left|\Delta \right|})\). We check our formula against the known numerical data arising from the Igusa cusp form, for the first 1650 polar coefficients, and find perfect agreement.A direct proof that toric rank \(2\) bundles on projective space split.https://www.zbmath.org/1456.140542021-04-16T16:22:00+00:00"Stapleton, David"https://www.zbmath.org/authors/?q=ai:stapleton.david-pSummary: The point of this paper is to give a short, direct proof that rank \(2\) toric vector bundles on \(n\)-dimensional projective space split once \(n\) is at least \(3\). This result is originally due to \textit{J. Bertin} and \textit{G. Elencwajg} [Duke Math. J. 49, 807--831 (1982; Zbl 0512.14007)], and there is also related work by \textit{T. Kaneyama} [Nagoya Math. J. 111, 25--40 (1988; Zbl 0820.14010)], \textit{A. A. Klyachko} [Math. USSR, Izv. 35, No. 2, 337--375 (1990; Zbl 0706.14010); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 5, 1001--1039 (1989)], and \textit{N. Ilten} and \textit{H. Süss} [Transform. Groups 20, No. 4, 1043--1073 (2015; Zbl 1387.14125)]. The idea is that, after possibly twisting the vector bundle, there is a section which is a complete intersection.Pseudoconvexity for the special Lagrangian potential equation.https://www.zbmath.org/1456.350892021-04-16T16:22:00+00:00"Harvey, F. Reese"https://www.zbmath.org/authors/?q=ai:harvey.reese"Lawson, H. Blaine jun."https://www.zbmath.org/authors/?q=ai:lawson.h-blaine-junSummary: The Special Lagrangian Potential Equation for a function \(u\) on a domain \(\Omega \subset \mathbb{R}^n\) is given by \(\mathrm{tr}\{\arctan (D^2 u)\} =\theta\) for a contant \(\theta \in (-n \frac{\pi}{2}, n \frac{\pi}{2})\). For \(C^2\) solutions the graph of \(Du\) in \(\Omega \times \mathbb{R}^n\) is a special Lagrangian submanfold. Much has been understood about the Dirichlet problem for this equation, but the existence result relies on explicitly computing the associated boundary conditions (or, otherwise said, computing the pseudo-convexity for the associated potential theory). This is done in this paper, and the answer is interesting. The result carries over to many related equations -- for example, those obtained by taking \(\sum_k \arctan \lambda_k^{\mathfrak{g}} =\theta\) where \(\mathfrak{g}: \mathrm{Sym}^2(\mathbb{R}^n)\rightarrow \mathbb{R}\) is a Gårding-Dirichlet polynomial which is hyperbolic with respect to the identity. A particular example of this is the deformed Hermitian-Yang-Mills equation which appears in mirror symmetry. Another example is \(\sum_j \arctan \kappa_j = \theta\) where \(\kappa_1,\dots, \kappa_n\) are the principal curvatures of the graph of \(u\) in \(\Omega \times \mathbb{R}\). We also discuss the inhomogeneous Dirichlet Problem
\[
\mathrm{tr}\{\arctan (D^2_x \,u)\} = \psi (x)
\]
where \(\psi : {\overline{\Omega}} \rightarrow (-n \frac{\pi}{2}, n \frac{\pi}{2})\). This equation has the feature that the pull-back of \(\psi\) to the Lagrangian submanifold \(L\equiv\text{graph} (Du)\) is the phase function \(\theta\) of the tangent spaces of \(L\). On \(L\) it satisfies the equation \(\nabla \psi = -JH\) where \(H\) is the mean curvature vector field of \(L\).Categorical mirror symmetry on cohomology for a complex genus 2 curve.https://www.zbmath.org/1456.530702021-04-16T16:22:00+00:00"Cannizzo, Catherine"https://www.zbmath.org/authors/?q=ai:cannizzo.catherineSummary: Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs \(X\) and \(Y\) such that the complex geometry on \(X\) mirrors the symplectic geometry on \(Y\). It allows one to deduce symplectic information about \(Y\) from known complex properties of \(X\). \textit{A. Strominger} et al. [Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)] described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. \textit{M. Kontsevich} [in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)] conjectured that a complex invariant on \(X\) (the bounded derived category of coherent sheaves) should be equivalent to a symplectic invariant of \(Y\) (the Fukaya category, see [\textit{D. Auroux}, Bolyai Soc. Math. Stud. 26, 85--136 (2014; Zbl 1325.53001); \textit{K. Fukaya} et al., Lagrangian intersection Floer theory. Anomaly and obstruction. I. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2009; Zbl 1181.53002); \textit{D. McDuff} et al., Virtual fundamental cycles in symplectic topology. New York, NY: American Mathematical Society (2019)]). This is known as homological mirror symmetry. In this project, we first use the construction of ``generalized SYZ mirrors'' for hypersurfaces in toric varieties following \textit{M. Abouzaid} et al. [Publ. Math., Inst. Hautes Étud. Sci. 123, 199--282 (2016; Zbl 1368.14056)], in order to obtain \(X\) and \(Y\) as manifolds. The complex manifold is the genus 2 curve \(\Sigma_2\) (so of general type \(c_1 < 0\)) as a hypersurface in its Jacobian torus. Its generalized SYZ mirror is a Landau-Ginzburg model \((Y, v_0)\) equipped with a holomorphic function \(v_0 : Y \to \mathbb{C}\) which we put the structure of a symplectic fibration on. We then describe an embedding of a full subcategory of \(D^b Coh(\Sigma_2)\) into a cohomological Fukaya-Seidel category of \(Y\) as a symplectic fibration. While our fibration is one of the first nonexact, non-Lefschetz fibrations to be equipped with a Fukaya category, the main geometric idea in defining it is the same as in Seidel's construction for Fukaya categories of Lefschetz fibrations in [\textit{P. Seidel}, Fukaya categories and Picard-Lefschetz theory. Zürich: European Mathematical Society (EMS) (2008; Zbl 1159.53001); \textit{M. Abouzaid} and \textit{P. Seidel}, ``Lefschetz fibration methods in wrapped Floer cohomology'', in preparation].Non-simply-connected symmetries in 6D SCFTs.https://www.zbmath.org/1456.814272021-04-16T16:22:00+00:00"Dierigl, Markus"https://www.zbmath.org/authors/?q=ai:dierigl.markus"Oehlmann, Paul-Konstantin"https://www.zbmath.org/authors/?q=ai:oehlmann.paul-konstantin"Ruehle, Fabian"https://www.zbmath.org/authors/?q=ai:ruehle.fabianSummary: Six-dimensional \(\mathcal{N} = (1, 0)\) superconformal field theories can be engineered geometrically via F-theory on elliptically-fibered Calabi-Yau 3-folds. We include torsional sections in the geometry, which lead to a finite Mordell-Weil group. This allows us to identify the full non-abelian group structure rather than just the algebra. The presence of torsion also modifies the center of the symmetry groups and the matter representations that can appear. This in turn affects the tensor branch of these theories. We analyze this change for a large class of superconformal theories with torsion and explicitly construct their tensor branches. Finally, we elaborate on the connection to the dual heterotic and M-theory description, in which our configurations are interpreted as generalizations of discrete holonomy instantons.A string theory realization of special unitary quivers in 3 dimensions.https://www.zbmath.org/1456.830942021-04-16T16:22:00+00:00"Collinucci, Andrés"https://www.zbmath.org/authors/?q=ai:collinucci.andres"Valandro, Roberto"https://www.zbmath.org/authors/?q=ai:valandro.robertoSummary: We propose a string theory realization of three-dimensional \(\mathcal{N} = 4\) quiver gauge theories with special unitary gauge groups. This is most easily understood in type IIA string theory with D4-branes wrapped on holomorphic curves in local K3's, by invoking the Stückelberg mechanism. From the type IIB perspective, this is understood as simply compactifying the familiar Hanany-Witten (HW) constructions on a \(T^3\). The mirror symmetry duals are easily derived. We illustrate this with various examples of mirror pairs.Moderately ramified actions in positive characteristic.https://www.zbmath.org/1456.140062021-04-16T16:22:00+00:00"Lorenzini, Dino"https://www.zbmath.org/authors/?q=ai:lorenzini.dino-j"Schröer, Stefan"https://www.zbmath.org/authors/?q=ai:schroer.stefanSummary: In characteristic 2 and dimension 2, wild \(\mathbb{Z}/2\mathbb{Z} \)-actions on \(k[[u, v]]\) ramified precisely at the origin were classified by \textit{M. Artin} [Proc. Am. Math. Soc. 52, 60--64 (1975; Zbl 0315.14015)], who showed in particular that they induce hypersurface singularities. We introduce in this article a new class of wild quotient singularities in any characteristic \(p>0\) and dimension \(n\ge 2\) arising from certain non-linear actions of \(\mathbb{Z}/p\mathbb{Z}\) on the formal power series ring \(k[[u_1,\dots,u_n]]\). These actions are ramified precisely at the origin, and their rings of invariants in dimension 2 are hypersurface singularities, with an equation of a form similar to the form found by Artin when \(p=2\). In higher dimension, the rings of invariants are not local complete intersection in general, but remain quasi-Gorenstein. We establish several structure results for such actions and their corresponding rings of invariants.An exact formula for \(\mathbf{U}(3)\) Vafa-Witten invariants on \(\mathbb{P}^2\).https://www.zbmath.org/1456.110752021-04-16T16:22:00+00:00"Bringmann, Kathrin"https://www.zbmath.org/authors/?q=ai:bringmann.kathrin"Nazaroglu, Caner"https://www.zbmath.org/authors/?q=ai:nazaroglu.canerSummary: Topologically twisted \(\mathcal{N} = 4\) super Yang-Mills theory has a partition function that counts Euler numbers of instanton moduli spaces. On the manifold \(\mathbb{P}^2\) and with gauge group \(\mathrm{U}(3)\) this partition function has a holomorphic anomaly which makes it a mock modular form of depth two. We employ the circle method to find a Rademacher expansion for the Fourier coefficients of this partition function. This is the first example of the use of circle method for a mock modular form of a higher depth.Homogeneous vector bundles over abelian varieties via representation theory.https://www.zbmath.org/1456.140522021-04-16T16:22:00+00:00"Brion, Michel"https://www.zbmath.org/authors/?q=ai:brion.michelLet \(A\) be an abelian variety over a field. A vector bundle on \(A\) is called homogeneous if it is invariant under pullback by translations on \(A\). Such bundles have been studied by many people, especially in the case when \(k\) is algebraically closed. The author proposes an alternative aproach to studying such bundles that works over an arbitrary field. This is based on the equivalence of the category of homogeoenus bundles with the category of finite-dimensional representations of a certain commutative affine \(k\)-group scheme \(H_A\), which plays a role of the affine fundamental group scheme of \(A\). Roughly speaking, by taking the inverse limit over all extensions \(1\to H\to G\to A\to 1\) with \(H\) affine, one arrives at the universal extension \(1\to H_A\to G_A\to A\to 1\). Then to a finite-dimensional \(H_A\)-module one can associate a homogeneous vector bundle \(G_A\times ^{H_A}V\to G_A/H_A=A\). Over an algebraically closed field \(k\) the group scheme \(H_A\) coincides with the S-fundamental group scheme studied by the reviewer.
The paper contains also various interesting results pertaining to representation theory of a commutative group scheme over a possibly imperfect field.
Reviewer: Adrian Langer (Warszawa)Picard-Vessiot groups of Lauricella's hypergeometric systems \(E_C\) and Calabi-Yau varieties arising integral representations.https://www.zbmath.org/1456.140142021-04-16T16:22:00+00:00"Goto, Yoshiaki"https://www.zbmath.org/authors/?q=ai:goto.yoshiaki"Koike, Kenji"https://www.zbmath.org/authors/?q=ai:koike.kenjiThe authors study the Zariski closure of the monodromy group \(Mon\) of Lauricella's hypergeometric function \(F_C(a,b,c;x)=\sum_{m_1,\ldots ,m_n=0}^{\infty}\frac{(a)_{m_1+\cdots +m_n}(b)_{m_1+\cdots +m_n}}{(c_1)_{m_1}\cdots (c_n)_{m_n}m_1!\cdots m_n!}x_1^{m_1}\cdots x_n^{m_n}\), where \(a,b\in \mathbb{C}\), \(c_i\in \mathbb{C}\setminus \{ 0,-1,-2,\ldots \}\), \((c_i)_{m_i}=\Gamma (c_i+m_i)/\Gamma (c_i)\), and Calabi-Yau varieties arising from its integral representation. When the identity component of \(Mon\) acts irreducibly, then \(\overline{Mon}\cap SL_{2^n}(\mathbb{C})\) is one of the classical groups \(SL_{2^n}(\mathbb{C})\), \(SO_{2^n}(\mathbb{C})\) or \(Sp_{2^n}(\mathbb{C})\).
Reviewer: Vladimir P. Kostov (Nice)M-theory and orientifolds.https://www.zbmath.org/1456.140462021-04-16T16:22:00+00:00"Braun, Andreas P."https://www.zbmath.org/authors/?q=ai:braun.andreas-pSummary: We construct the M-Theory lifts of type IIA orientifolds based on \(K3\)-fibred Calabi-Yau threefolds with compatible involutions. Such orientifolds are shown to lift to M-Theory on twisted connected sum \(G_2\) manifolds. Beautifully, the two building blocks forming the \(G_2\) manifold correspond to the open and closed string sectors. As an application, we show how to use such lifts to explicitly study open string moduli. Finally, we use our analysis to construct examples of \(G_2\) manifolds with different inequivalent TCS realizations.