Deformations of Toric Varieties via Minkowski Sum Decompositions of Polyhedral Complexes

TL;DR

This paper introduces a new method for deforming arbitrary toric varieties using Minkowski sum decompositions of polyhedral complexes, generalizing previous constructions and connecting to mirror symmetry.

Contribution

It extends deformation constructions to all toric varieties via Minkowski sums, embedding into higher-dimensional toric varieties and linking to mirror symmetry frameworks.

Findings

Deformations span the infinitesimal deformation space for certain Gorenstein toric varieties.

Constructs deformations as families of complete intersections in higher-dimensional toric varieties.

Connects deformations of Fano toric varieties to Batyrev-Borisov mirror symmetry.

Abstract

We generalized the construction of deformations of affine toric varieties of K. Altmann and our previous construction of deformations of weak Fano toric varieties to the case of arbitrary toric varieties by introducing the notion of Minkowski sum decompositions of polyhedral complexes. Our construction embeds the original toric variety into a higher dimensional toric variety where the image is given by a prime binomial complete intersection ideal in Cox homogeneous coordinates. The deformations are realized by families of complete intersections. For compact simplicial toric varieties with at worst Gorenstein terminal singularities, we show that our deformations span the infinitesimal space of deformations by Kodaira-Spencer map. For Fano toric varieties, we show that their deformations can be constructed in higher-dimensional Fano toric varieties related to the Batyrev-Borisov mirror symmetry construction.