Intersection pairings on singular moduli spaces of bundles over a Riemann surface and their partial desingularisations

Abstract

This paper studies intersection theory on the compactified moduli space ${\mbox{$\cal M$}} (n,d)$ of holomorphic bundles of rank n and degree d over a fixed compact Riemann surface $\Sigma$ of genus $g \geq 2$ where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups $I\!H^*({\mbox{$\cal M$}} (n,d))$ defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities $\widetilde{{\mbox{$\cal M$}}} (n,d)$ of ${\mbox{$\cal M$}} (n,d).$ Based on our earlier work [25], we give a precise formula for the intersection cohomology pairings and provide a method to calculate pairings on $\widetilde{{\mbox{$\cal M$}}}(n,d).$ The case when n = 2 is discussed in detail. Finally Witten's integral is considered for this singular case.