Fixed point loci of moduli spaces of sheaves on toric varieties

TL;DR

This paper develops a combinatorial framework for describing equivariant sheaves on toric varieties, constructs explicit moduli spaces using GIT, and analyzes fixed point loci to facilitate computations of invariants like Euler characteristics.

Contribution

It introduces a combinatorial description of pure equivariant sheaves on nonsingular toric varieties and constructs explicit moduli spaces using GIT, extending prior work.

Findings

Explicit description of fixed point loci in moduli spaces

Construction of moduli spaces of pure equivariant sheaves

Application to computing Euler characteristics of sheaf moduli spaces

Abstract

Extending work of Klyachko and Perling, we develop a combinatorial description of pure equivariant sheaves of any dimension on an arbitrary nonsingular toric variety $X$. Using geometric invariant theory (GIT), this allows us to construct explicit moduli spaces of pure equivariant sheaves on $X$ corepresenting natural moduli functors (similar to work of Payne in the case of equivariant vector bundles). The action of the algebraic torus on $X$ lifts to the moduli space of all Gieseker stable sheaves on $X$ and we express its fixed point locus explicitly in terms of moduli spaces of pure equivariant sheaves on $X$. One of the problems arising is to find an equivariant line bundle on the side of the GIT problem, which precisely recovers Gieseker stability. In the case of torsion free equivariant sheaves, we can always construct such equivariant line bundles. As a by-product, we get a combinatorial description of the fixed point locus of the moduli space of $\mu$-stable reflexive sheaves on $X$. As an application, we show in a sequel how these methods can be used to compute generating functions of Euler characteristics of moduli spaces of $\mu$-stable torsion free sheaves on nonsingular complete toric surfaces.