arXiv subjects

Victor Batyrev

Publications and source records attributed to Victor Batyrev.

22 records · Page 2Linked to original sources

arXiv2007

Combinatorial aspects of mirror symmetry

The purpose of this paper is to review some combinatorial ideas behind the mirror symmetry for Calabi-Yau hypersurfaces and complete intersections in Gorenstein toric Fano varieties. We suggest as a basic combinatorial object the notion of a Gorenstein polytope of index r. A natural combinatorial duality for d-dimensional Gorenstein polytopes of index r extends the well-known polar duality for reflexive polytopes (case r=1). We consider the Borisov duality between two nef-partitions as a duality between two Gorenstein polytopes P and P^* of index r together with selected special (r-1)-dimensional simplices S in P and S' in P^*. Different choices of these simplices suggest an interesting relation to Homological Mirror Symmetry.

math.CO↗

arXiv2006

Multiples of lattice polytopes without interior lattice points

Let $Δ$ be an $n$-dimensional lattice polytope. The smallest non-negative integer $i$ such that $k Δ$ contains no interior lattice points for $1 \leq k \leq n - i$ we call the degree of $Δ$. We consider lattice polytopes of fixed degree $d$ and arbitrary dimension $n$. Our main result is a complete classification of $n$-dimensional lattice polytopes of degree $d=1$. This is a generalization of the classification of lattice polygons $(n=2)$ without interior lattice points due to Arkinstall, Khovanskii, Koelman and Schicho. Our classification shows that the secondary polytope of a lattice polytope of degree 1 is always a simple polytope.

math.CO↗

arXiv2006

Lattice polytopes with a given $h^*$-polynomial

Let $Δ\subset \R^n$ be an $n$-dimensional lattice polytope. It is well-known that $h_Δ^*(t) := (1-t)^{n+1} \sum_{k \geq 0} |kΔ\cap \Z^n| t^k $ is a polynomial of degree $d \leq n$ with nonnegative integral coefficients. Let $AGL(n, \Z)$ be the group of invertible affine integral transformations which naturally acts on $\R^n$. For a given polynomial $h^* \in \Z[t]$, we denote by $C_{h^*}(n)$ the number $AGL(n, \Z)$-equivalence classes of $n$-dimensional lattice polytopes such that $h^* = h_Δ^*(t)$. In this paper we show that $\{C_{h^*}(n) \}_{n \geq 1}$ is a monotone increasing sequence which eventually becomes constant. This statement follows from a more general combinatorial result whose proof uses methods of commutative algebra.

math.CO↗

arXiv2005

Integral Cohomology and Mirror Symmetry for Calabi-Yau 3-folds

In this paper, we compute the integral cohomology groups for all examples of Calabi-Yau 3-folds obtained from hypersurfaces in 4-dimensional Gorenstein toric Fano varieties. Among 473 800 776 families of Calabi-Yau 3-folds $X$ corresponding to 4-dimensional reflexive polytopes there exist exactly 32 families having non-trivial torsion in $H^*(X, \Z)$. We came to an interesting observation that the torsion subgroups in $H^2$ and $H^3$ are exchanged by the mirror symmetry involution, i.e. the torsion subgroup in the Picard group of $X$ is isomorphic to the Brauer group of the mirror $X^*$

math.AG↗