Deformations of Rational T-Varieties

TL;DR

This paper develops a method to construct homogeneous deformations of rational T-varieties and toric varieties, analyzing their deformation spaces and the role of Kodaira-Spencer maps in these deformations.

Contribution

It introduces a construction for homogeneous deformations of rational T-varieties and computes the Kodaira-Spencer map for these deformations, showing their sufficiency for smooth complete toric varieties.

Findings

Homogeneous deformations can be explicitly constructed for rational T-varieties.

The Kodaira-Spencer map image is computed for locally trivial deformations.

Homogeneous deformations span the entire first-order deformation space for smooth complete toric varieties.

Abstract

We show how to construct certain homogeneous deformations for rational normal varieties with codimension one torus action. This can then be used to construct homogeneous deformations of any toric variety in arbitrary degree. For locally trivial deformations coming from this construction, we calculate the image of the Kodaira-Spencer map. We then show that for a smooth complete toric variety, our homogeneous deformations span the space of first-order deformations.