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

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.