Sheaves on weighted projective planes and modular forms

TL;DR

This paper provides an explicit description of toric sheaves on weighted projective planes viewed as stacks, computes fixed point loci of moduli schemes of sheaves, and connects generating functions to modular forms, generalizing previous results.

Contribution

It introduces a detailed description of toric sheaves on weighted projective planes as stacks, including gerbe and root structures, and derives explicit formulas for generating functions related to moduli spaces.

Findings

Explicit formulas for generating functions of Euler characteristics.

Connection of generating functions to modular forms of weight 3/2.

Generalization of Klyachko's results on $\

Abstract

We give an explicit description of toric sheaves on the weighted projective plane $\mathbb{P}(a,b,c)$ viewed as a toric Deligne-Mumford stack. The integers $(a,b,c)$ are not necessarily chosen coprime or mutually coprime allowing for gerbe and root stack structures. As an application, we describe the fixed point locus of the moduli scheme of stable rank 1 and 2 torsion free sheaves on $\mathbb{P}(a,b,c)$ with fixed $K$-group class. Summing over all $K$-group classes, we obtain explicit formulae for generating functions of the topological Euler characteristics. In the case of stable rank 2 locally free sheaves on $\mathbb{P}(a,b,c)$ with $a,b,c \leq 2$ the generating functions can be expressed in terms of Hurwitz class numbers and give rise to modular forms of weight $3/2$. This generalizes Klyachko's computation on $\mathbb{P}^2$ and is consistent with $S$-duality predictions from physics.