On deformations of toric varieties

TL;DR

This paper characterizes equivariant deformations of smooth complete toric varieties using Cox rings, constructing explicit families and relating them to the H1 cohomology group.

Contribution

It provides a Cox ring-based description of equivariant deformations and constructs explicit one-parameter families with total spaces as T-varieties.

Findings

Constructed explicit deformation families as T-varieties of complexity one

Showed these families form a basis of H1(X,T_X) via the Kodaira-Spencer map

Connected deformation theory with Cox ring and T-variety structures

Abstract

Let X be a smooth complete toric variety. We describe the Altmann-Ilten-Vollmert equivariant deformations of toric varieties in the language of Cox rings. More precisely we construct one parameters families of deformations of X, such that the total space of the deformation is a T-variety of complexity one defined by a trinomial equation, and the deformation map is equivariant with respect to the torus action. Moreover we show that the images of all these families via the Kodaira-Spencer map form a basis of the vector space H1(X,T_X).