Intersection theory on singular moduli spaces of vector bundles: a parabolic approach

TL;DR

This paper develops explicit formulas for intersection pairings in the intersection cohomology of singular moduli spaces of semistable vector bundles on Riemann surfaces, using a parabolic bundle approach and residue formulas.

Contribution

It introduces a novel parabolic bundle method to compute intersection pairings, simplifying previous blow-up techniques and providing clear geometric interpretations.

Findings

Explicit formulas for intersection pairings on $M_0(r)$ for all $r$

Reduction of complex calculations to residue formulas via parabolic bundles

Simplification over previous blow-up constructions

Abstract

We present explicit formulas for the intersection pairing in the intersection cohomology of the moduli space $M_0(r)$ of rank-$r$, degree-$0$ semistable bundles on a Riemann surface. The key idea is to realize this intersection cohomology as a canonical subspace of the cohomology of a smooth moduli space of parabolic bundles, where the pairing can be computed via the Hecke correspondence and the Jeffrey-Kirwan iterated residue formulas. This approach provides a simpler alternative to the blow-up construction of Jeffrey-Kirwan-Kiem-Woolf, yielding formulas for the intersection pairing on $M_0(r)$, for arbitrary $r$, with a clear geometric interpretation.