Fixed point loci of moduli spaces of sheaves on toric varieties

TL;DR

The work develops a comprehensive combinatorial framework for pure equivariant sheaves on nonsingular toric varieties via $\sigma$-families, enabling explicit GIT constructions of moduli spaces that corepresent natural moduli functors. It then identifies and describes the fixed point loci of moduli spaces of Gieseker stable (and μ-stable reflexive) sheaves under the torus action as disjoint unions of moduli spaces of pure equivariant (torsion-free or reflexive) sheaves, with a precise bridge between Gieseker and GIT stability through ample equivariant line bundles. The results provide concrete, computable descriptions of $\left(\mathcal{M}_{P}^{s}\right)^{T}$ and $\left(\mathcal{N}_{P}^{\mu s}\right)^{T}$ in terms of framed $\Delta$-families, enabling localization-based computations of invariants such as Euler characteristics and informing wall-crossing phenomena on toric surfaces. The framework also yields explicit formulas for Chern characters (via Klyachko’s formula) and Hilbert polynomials from combinatorial data, tying toric geometry directly to moduli theory of coherent sheaves with practical computational implications.

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 $μ$-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 $μ$-stable torsion free sheaves on nonsingular complete toric surfaces.

Theorems & Definitions (81)

• Theorem 1.1: Corollary \ref{['ch. 1, sect. 4, cor. 1']}
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Definition 2.6