Counting sheaves on curves

Abstract

We compute Joyce's [14] enumerative invariants [M(r,d)ss]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 [23] and proved by Jeffrey and Kirwan [12]. 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.