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Victor V. Batyrev

Publications and source records attributed to Victor V. Batyrev.

At least 19 recordsLinked to original sources
arXiv2023

Hodge Theory of Hypersurfaces in Toric Varieties and Recent Developments in Quantum Physics

This is the author's Habilitation which took place at University of Essen on July 11, 1993. The manuscript contains two parts. The first one is devoted to the author's combinatorial construction of mirrors of Calabi-Yau hypersurfaces in Gorenstein toric Fano varieties. The second one contains author's results on the variation of mixed Hodge structures of affine hypersurfaces in algebraic tori and their connection to Gelfand-Kapranov-Zelevinsky theory of generalized hypergeometric functions and their applications to the mirror symmetry for Calabi-Yau hypersurfaces in toric varieties.

math.AG↗
arXiv2023

Projecting lattice polytopes according to the Minimal Model Program

The Fine interior $F(P)$ of a $d$-dimensional lattice polytope $P \subset {\Bbb R}^d$ is the set of all points $y \in P$ having integral distance at least $1$ to any integral supporting hyperplane of $P$. We call a lattice polytope $F$-hollow if its Fine interior is empty. The main theorem claims that up to unimodular equivalence in each dimension $d$ there exist only finitely many $d$-dimensional $F$-hollow lattice polytopes $P$, so called {\em sporadic}, which do not admit a lattice projection onto a $k$-dimensional $F$-hollow lattice polytope $P'$ for some $1 \leq k \leq d-1$. The proof is purely combinatorial, but it is inspired by ${\Bbb Q}$-Fano fibrations in the Minimal Model Program, since we show that non-degenerate toric hypersurfaces $Z \subset ({\Bbb C}^*)^d$ defined by zeros of Laurent polynomials with a given Newton polytope $P$ have negative Kodaira dimension if and only if $P$ is $F$-hollow. The finiteness theorem for $d$-dimensional sporadic $F$-hollow Newton polytopes $P$ gives rise to finitely many families ${\mathcal F}(P)$ of $(d-1)$-dimensional ${\Bbb Q}$-Fano hypersurfaces with at worst canonical singularities.

math.AG↗
arXiv2023

Canonical models of toric hypersurfaces

Let $Z$ be a nondegenerate hypersurface in $d$-dimensional torus $(\mathbb{C}^*)^d$ defined by a Laurent polynomial $f$ with a $d$-dimensional Newton polytope $P$. The subset $F(P) \subset P$ consisting of all points in $P$ having integral distance at least $1$ to all integral supporting hyperplanes of $P$ is called the Fine interior of $P$. If $F(P) \neq \emptyset$ we construct a unique projective model $\widetilde{Z}$ of $Z$ having at worst canonical singularities and obtain minimal models $\hat{Z}$ of $Z$ by crepant morphisms $\hat{Z}\to \widetilde{Z}$. We show that the Kodaira dimension $κ=κ(\widetilde{Z})$ equals $\min \{ d-1, \dim F(P) \}$ and the general fibers in the Iitaka fibration of the canonical model $\widetilde{Z}$ are non\-degenerate $(d-1-κ)$-dimensional toric hypersurfaces of Kodaira dimension $0$. Using $F(P)$, we obtain a simple combinatorial formula for the intersection number $(K_{\widetilde{Z}})^{d-1}$.

math.AG↗
arXiv2020

On the stringy Hodge numbers of mirrors of quasi-smooth Calabi-Yau hypersurfaces

Mirrors $X^{\vee}$ of quasi-smooth Calabi-Yau hypersurfaces $X$ in weighted projective spaces ${\Bbb P}(w_0, \ldots, w_d)$ can be obtained as Calabi-Yau compactifications of non-degenerate affine toric hypersurfaces defined by Laurent polynomials whose Newton polytope is the lattice simplex spanned by $d+1$ lattice vectors $v_i$ satisfying the relation $\sum_i w_i v_i =0$. In this paper, we compute the stringy $E$-function of mirrors $X^\vee$ and compare it with the Vafa's orbifold $E$-function of quasi-smooth Calabi-Yau hypersurfaces $X$. As a result, we prove the equalities of Hodge numbers $h^{p,q}_{\rm str}(X^{\vee}) = h^{d-1-p,q}_{\rm orb}(X)$ for all $p, q$ and $d$ as it is expected in mirror symmetry.

math.AG↗
arXiv2003

The Cox ring of a Del Pezzo surface

Let $X_r$ be a smooth Del Pezzo surface obtained from $¶^2$ by blowing up $r \leq 8$ points in general position. It is well known that for $r \in \{3,4,5,6,7,8 \}$ the Picard group $\Pic(X_r)$ contains a canonical root system $R_r \in \{A_2 \times A_1, A_4, D_5, E_6, E_7, E_8 \}$. We prove some general properties of the Cox ring of $X_r$ ($r \geq 4$) and show its similarity to the homogeneous coordinate ring of the orbit of the highest weight vector in some irreducible representation of the algebraic group $G$ associated with the root system $R_r$.

math.AG↗
arXiv2003

Mixed toric residues and Calabi-Yau complete intersections

Using Cayley trick, we define the notions of mixed toric residues and mixed Hessians associated with $r$ Laurent polynomials $f_1,...,f_r$.We conjecture that the values of mixed toric residues on the mixed Hessians are determined by mixed volumes of the Newton polytopes of $f_1,...,f_r$. Using mixed toric residues, we generalize our Toric Residue Mirror Conjecture to the case of Calabi-Yau complete intersections in Gorenstein toric Fano varieties obtained from nef-partitions of reflexive polytopes.

math.AG↗
arXiv2002

Toric Residues and Mirror Symmetry

We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture which has close relations to toric mirror symmetry. Our conjecture, we call it Toric Residue Mirror Conjecture, claims that the generating functions of intersection numbers of divisors on a special sequence of simplicial toric varieties are power series expansions of some rational functions obtained as toric residues. We expect that this conjecture holds true for all Gorenstein toric Fano varieties associated with reflexive polytopes and give some evidences for that. The proposed conjecture suggests a simple method for computing Yukawa couplings for toric mirror Calabi-Yau hypersurfaces without solving systems of differential equations. We make several explicit computations for Calabi-Yau hypersurfaces in weighted projective spaces and in products of projective spaces.

math.AG↗
arXiv1999

Einstein-Kaehler Metrics on Symmetric Toric Fano Manifolds

Let $X$ be a complex toric Fano $n$-fold and ${\cal N}(T)$ the normalizer of a maximal torus $T$ in the group of biholomorphic authomorphisms $Aut(X)$. We call $X$ {\em symmetric} if the trivial character is a single ${\cal N}(T)$-invariant algebraic character of $T$. Using an invariant $α_G(X)$ introduced by Tian, we show that all symmetric toric Fano $n$-folds admit an Einstein-Kähler metric. We remark that so far one doesn't know any example of a toric Fano $n$-fold $X$ such that $Aut(X)$ is reductive, the Futaki character of $X$ vanishes, but $X$ is not symmetric.

math.AG↗
arXiv1998

Mirror Symmetry and Toric Degenerations of Partial Flag Manifolds

In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds $F(n_1, ..., n_l, n)$. This construction includes our previous mirror construction for complete intersection in Grassmannians and the mirror construction of Givental for complete flag manifolds. The key idea of our construction is a degeneration of $F(n_1, ..., n_l, n)$ to a certain Gorenstein toric Fano variety $P(n_1, ..., n_l, n)$ which has been investigated by Gonciulea and Lakshmibai. We describe a natural small crepant desingularization of $P(n_1, ..., n_l, n)$ and prove a generalized version of a conjecture of Gonciulea and Lakshmibai on the singular locus of $P(n_1, ..., n_l, n)$.

math.AG↗
arXiv1998

Birational Calabi--Yau n-folds have equal Betti numbers

Let X and Y be two smooth projective n-dimensional algebraic varieties X and Y over C with trivial canonical line bundles. We use methods of p-adic analysis on algebraic varieties over local number fields to prove that if X and Y are birational, they have the same Betti numbers.

alg-geom↗
arXiv1998

Stringy Hodge numbers of varieties with Gorenstein canonical singularities

We introduce the notion of stringy E-function for an arbitrary normal irreducible algebraic variety X with at worst log-terminal singularities. We prove some basic properties of stringy E-functions and compute them explicitly for arbitrary Q-Gorenstein toric varieties. Using stringy E-functions, we propose a general method to define stringy Hodge numbers for projective algebraic varieties with at worst Gorenstein canonical singularities. This allows us to formulate the topological mirror duality test for arbitrary Calabi-Yau varieties with canonical singularities. In Appendix we explain non-Archimedian integrals over spaces of arcs. We need these integrals for the proof of the main technical statement used in the definition of stringy Hodge numbers.

alg-geom↗
arXiv1998

Non-Archimedian integrals and stringy Euler numbers of log terminal pairs

Using non-Archimedian integration over spaces of arcs of algebraic varieties, we define stringy Euler numbers associated with arbitrary Kawamata log terminal pairs. There is a natural Kawamata log terminal pair corresponding to an algebraic variety V having a regular action of a finite group G. In this situation we show that the stringy Euler number of this pair coincides with the physicists' orbifold Euler number defined by the Dixon-Harvey-Vafa-Witten formula. As an application, we prove a conjecture of Miles Reid on the Euler numbers of crepant desingularizations of Gorenstein quotient singularities.

math.AG↗
arXiv1998

On the Classification of Toric Fano 4-folds

In this paper we explain the complete biregular classification of all 4-dimensional smooth toric Fano varieties. The main result states that there exist exactly 123 different types of toric Fano 4-folds up to isomorphism.

math.AG↗
arXiv1997

Toric Degenerations of Fano Varieties and Constructing Mirror Manifolds

For an arbitrary smooth n-dimensional Fano variety $X$ we introduce the notion of a small toric degeneration. Using small toric degenerations of Fano n-folds $X$, we propose a general method for constructing mirrors of Calabi-Yau complete intersections in $X$. Our mirror construction is based on a generalized monomial-divisor mirror correspondence which can be used for computing Gromov-Witten invariants of rational curves via specializations of GKZ-hypergeometric series.

alg-geom↗
arXiv1997

Tamagawa numbers of polarized algebraic varieties

Let ${\cal L} = (L, \| \cdot \|_v)$ be an ample metrized invertible sheaf on a smooth quasi-projective algebraic variety $V$ defined over a number field. Denote by $N(V,{\cal L},B)$ the number of rational points in $V$ having ${\cal L}$-height $\leq B$. We consider the problem of a geometric and arithmetic interpretation of the asymptotic for $N(V,{\cal L},B)$ as $B \to \infty$ in connection with recent conjectures of Fujita concerning the Minimal Model Program for polarized algebraic varieties. We introduce the notions of ${\cal L}$-primitive varieties and ${\cal L}$-primitive fibrations. For ${\cal L}$-primitive varieties $V$ over $F$ we propose a method to define an adelic Tamagawa number $τ_{\cal L}(V)$ which is a generalization of the Tamagawa number $τ(V)$ introduced by Peyre for smooth Fano varieties. Our method allows us to construct Tamagawa numbers for $Q$-Fano varieties with at worst canonical singularities. In a series of examples of smooth polarized varieties and singular Fano varieties we show that our Tamagawa numbers express the dependence of the asymptotic of $N(V,{\cal L},B)$ on the choice of $v$-adic metrics on ${\cal L}$.

alg-geom↗
arXiv1997

Stringy Hodge numbers and Virasoro algebra

Let $X$ be an arbitrary smooth $n$-dimensional projective variety. It was discovered by Libgober and Wood that the product of the Chern classes $c_1(X)c_{n-1}(X)$ depends only on the Hodge numbers of $X$. This result has been used by Eguchi, Jinzenji and Xiong in their approach to the quantum cohomology of $X$ via a representation of the Virasoro algebra with the central charge $c_n(X)$. In this paper we define for singular varieties $X$ a rational number $c_{st}^{1,n-1}(X)$ which is a stringy version of the number $c_1c_{n-1}$ for smooth $n$-folds. We show that the number $c_{st}^{1,n-1}(X)$ can be expressed in the same way using the stringy Hodge numbers of $X$. Our results provides an evidence for the existence of an approach to quantum cohomology of singular varieties $X$ via a representation of the Virasoro algebra whose central charge is the rational number $e_{st}(X)$ which equals the stringy Euler number of $X$.

alg-geom↗
arXiv1997

Conifold Transitions and Mirror Symmetry for Calabi-Yau Complete Intersections in Grassmannians

In this paper we show that conifold transitions between Calabi-Yau 3-folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection Calabi-Yau 3-folds in Grassmannians. Using a natural degeneration of Grassmannians $G(k,n)$ to some Gorenstein toric Fano varieties $P(k,n)$ with conifolds singularities which was recently described by Sturmfels, we suggest an explicit mirror construction for Calabi-Yau complete intersections $X \subset G(k,n)$ of arbitrary dimension. Our mirror construction is consistent with the formula for the Lax operator conjectured by Eguchi, Hori and Xiong for gravitational quantum cohomology of Grassmannians.

alg-geom↗