Counting sheaves on curves

TL;DR

This paper computes Joyce's enumerative invariants for semistable sheaves on curves, providing explicit formulas and a new perspective on classical cohomology pairings, especially when rank and degree are not coprime.

Contribution

It introduces a method to compute Joyce's invariants for all semistable sheaves on curves, extending classical results to the non-coprime case with explicit formulas.

Findings

Explicit formulas for Joyce's invariants on curves.

A regularized sum approach to divergent series.

Extension of classical cohomology pairing results to non-coprime cases.

Abstract

We compute Joyce's (arXiv:2111.04694) enumerative invariants $[\mathcal{M}^{\mathrm{ss}}_{(r,d)}]_{\mathrm{inv}}$ for semistable rank $r$ degree $d$ coherent sheaves on a complex projective curve. These invariants are a generalization of the fundamental class of the moduli of semistable sheaves. We express the invariants as a regularized sum, which is a way to assign finite values to divergent series, and we obtain explicit expressions for the invariants. From these invariants, one can extract cohomology pairings on the moduli of semistable sheaves. When $r$ and $d$ are coprime, formulae for such pairings were found by Witten and proved by Jeffrey and Kirwan. Our results provide a new point of view on this classical problem, and can be seen as a generalization of this to the case when $r$ and $d$ are not coprime.