Equivariant Vector Bundles on T-Varieties

TL;DR

This paper generalizes the classification of equivariant vector bundles from toric varieties to T-varieties, providing a categorical equivalence under factoriality and applications to splitting theorems and vector fields.

Contribution

It introduces a functorial framework for equivariant vector bundles on T-varieties, extending Klyachko's work and establishing new splitting and deformation results.

Findings

Categorical equivalence for factorial T-varieties

Splitting of low-rank equivariant bundles on projective space

Descriptions of global vector fields and deformations

Abstract

Let X be a T-variety, where T is an algebraic torus. We describe a fully faithful functor from the category of T-equivariant vector bundles on X to a certain category of filtered vector bundles on a suitable quotient of X by T. We show that if X is factorial, this functor gives an equivalence of categories. This generalizes Klyachko's description of equivariant vector bundles on toric varieties. We apply our machinery to show that vector bundles of low rank on projective space which are equivariant with respect to special subtori of the maximal torus must split, generalizing a theorem of Kaneyama. Further applications include descriptions of global vector fields on T-varieties, and a study of equivariant deformations of equivariant vector bundles.