Sheaves on weighted projective planes and modular forms

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.