Explanipedia

On the classification of smooth toric surfaces with exactly one exceptional curve Open

Victor V. Batyrev · 2024

We classify all smooth projective toric surfaces $S$ containing exactly one exceptional curve. We show that every such surface $S$ is isomorphic to either $\mathbb{F}_1$ or a surface $S_r$ defined by a rational number $r \in \mathbb{Q} \se…

Spherical amoebae and a spherical logarithm map Open

Victor V. Batyrev, Megumi Harada, Johannes Hofscheier, Kiumars Kaveh · 2024

Let $G$ be a connected reductive algebraic group over $\mathbb{C}$ with a maximal compact subgroup $K$. Let $G/H$ be a (quasi-affine) spherical homogeneous space. In the first part of the paper, following Akhiezer's definition of spherical…

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

Victor V. Batyrev · 2023

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 Go…

Projecting lattice polytopes according to the Minimal Model Program Open

Victor V. Batyrev · 2023

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 $F$-hol…

Canonical models of toric hypersurfaces Open

Victor V. Batyrev · 2023

Let Z be a nondegenerate hypersurface in a d-dimensional torus (C * ) d defined by a Laurent polynomial f with a d-dimensional Newton polytope P .The subset F (P ) ⊂ P consisting of all points in P having integral distance at least 1 to al…

On the Fine Interior of Three-Dimensional Canonical Fano Polytopes Open

Victor V. Batyrev, Alexander Kasprzyk, Karin Schaller · 2022

A generalization of a theorem of White Open

Victor V. Batyrev, Johannes Hofscheier · 2021

An m-dimensional simplex ∆ in Rm is called empty lattice simplex if ∆ ∩ Zm is exactly the set of vertices of ∆. A theorem of White states that if m = 3 then, up to an affine unimodular transformation of the lattice Zm, any empty lattice si…

Mirror symmetry for quasi-smooth Calabi–Yau hypersurfaces in weighted projective spaces Open

Victor V. Batyrev, Karin Schaller · 2021

Canonical models of toric hypersurfaces Open

Victor V. Batyrev · 2020

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…

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

Victor V. Batyrev · 2020

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 poly…

On the Fine Interior of Three-dimensional Canonical Fano Polytopes Open

Victor V. Batyrev, Alexander Kasprzyk, Karin Schaller · 2019

The Fine interior $Δ^{\text{FI}}$ of a $d$-dimensional lattice polytope $Δ$ is a rational subpolytope of $Δ$ which is important for constructing minimal birational models of non-degenerate hypersurfaces defined by Laurent polynomials with …

Stringy -functions of canonical toric Fano threefolds and their applications Open

Victor V. Batyrev, Karin Schaller · 2018

Let be a -dimensional lattice polytope containing exactly one interior lattice point. We give a simple combinatorial formula for computing the stringy -function of the -dimensional canonical toric Fano variety associated with . Using the s…

The stringy Euler number of Calabi-Yau hypersurfaces in toric varieties\n and the Mavlyutov duality Open

Victor V. Batyrev · 2017

We show that minimal models of nondegenerated hypersufaces defined by Laurent\npolynomials with a $d$-dimensional Newton polytope $\\Delta$ are Calabi-Yau\nvarieties $X$ if and only if the Fine interior of $\\Delta$ consists of a single\nl…

On the algebraic stringy Euler number Open

Victor V. Batyrev, Giuliano Gagliardi · 2017

We are interested in stringy invariants of singular projective algebraic varieties satisfying a strict monotonicity with respect to elementary birational modifications in the Mori program. We conjecture that the algebraic stringy Euler num…

Stringy Chern classes of singular toric varieties and their applications Open

Victor V. Batyrev, Karin Schaller · 2017

Let X be a normal projective Q-Gorenstein variety with at worst log-terminal singularities. We prove a formula expressing the total stringy Chern class of a generic complete intersection in X via the total stringy Chern class of X. This fo…

The stringy Euler number of Calabi–Yau hypersurfaces in toric varieties and the Mavlyutov duality Open

Victor V. Batyrev · 2017

We show that minimal models of nondegenerated hypersufaces defined by Laurent polynomials with a $d$-dimensional Newton polytope $Δ$ are Calabi-Yau varieties $X$ if and only if the Fine interior of $Δ$ consists of a single lattice point. W…

Satellites of spherical subgroups Open

Victor V. Batyrev, Anne Moreau · 2016

Let $G$ be a complex connected reductive algebraic group. Given a spherical subgroup $H \subset G$ and a subset $I$ of the set of spherical roots of $G/H$, we define, up to conjugation, a spherical subgroup $H_I \subset G$ of the same dime…

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