Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces

TL;DR

This paper derives a general formula for the Euler characteristics of moduli spaces of torsion free sheaves on smooth toric surfaces, expressed via stable configurations and applicable to various examples.

Contribution

It provides a new explicit combinatorial formula for the Euler characteristics of these moduli spaces, extending previous work to a broad class of toric surfaces and sheaves.

Findings

Derived a general generating function for Euler characteristics

Expressed the generating function in terms of stable configurations

Computed new and known examples of generating functions

Abstract

Given a smooth toric variety $X$, the action of the torus $T$ lifts to the moduli space $\mathcal{M}$ of stable sheaves on $X$. Using the pioneering work of Klyacho, a fairly explicit combinatorial description of the fixed point locus $\mathcal{M}^T$ can be given (as shown by earlier work of the author). In this paper, we apply this description to the case of torsion free sheaves on a smooth toric surface $S$. A general expression for the generating function of the Euler characteristics of such moduli spaces is obtained. The generating function is expressed in terms of Euler characteristics of certain moduli spaces of stable configurations of linear subspaces appearing in classical GIT. The expression holds for any choice of $S$, polarization, rank, and first Chern class. Specializing to various examples allows us to compute some new as well as known generating functions.