Vector bundles on toric stacks

Abstract

This thesis is concerned with a generalization of Klyachko's classification of toric vector bundles to toric stacks. The work of Klyachko gave an elegant method of studying toric vector bundles through filtrations of a vector space. We extend these techniques to vector bundles on a toric stack and generalize the aspects of Klyachko's work to a more geometric setting.

In particular, we show that the category of reflexive sheaves on a toric stack is equivalent to a category of filtered reflexives sheaves of its largest Deligne-Mumford substack. We then combine this with an equivariant version of Gubeladze's result on the splitting of vector bundles on toric varieties to prove a classification theorem for vector bundles on toric stacks. As an application we reprove a known result on the splitting of rank-$2$ bundles on $[\bP^n/\bG_m]$ for a particular $\bG_m$ action.

Our methods involve an extension of Cox's construction of homogeneous coordinates to toric stacks and we incorporate ideas from the classical Rees construction. We also study the Chow ring of toric stacks, and give a presentation of the Chow ring of a smooth toric stack.