Parabolic bundles and the intersection cohomology of moduli spaces of vector bundles on curves

TL;DR

This paper provides a geometric proof of a recursive formula for calculating intersection Betti numbers of moduli spaces of semistable bundles on Riemann surfaces, connecting known results for degree-1 to arbitrary degree.

Contribution

It introduces a new geometric proof of a recursive formula for intersection cohomology, utilizing the Decomposition Theorem and detailed topology of parabolic moduli spaces.

Findings

Recursive formula for intersection Betti numbers derived

Connection between degree-0 and degree-1 moduli spaces established

Detailed description of the topology of the forgetful map

Abstract

The study of the intersection cohomology of moduli spaces of semistable bundles was initiated by Frances Kirwan in the 1980's. In this paper, we give a complete geometric proof of a recursive formula, which reduces the calculation of the intersection Betti numbers of the moduli spaces of semistable bundles on Riemann surfaces in degree-0 and arbitrary rank to the known formulas of the Betti numbers of the smooth, degree-1 moduli spaces. Our formula was motivated by the work of Mozgovoy and Reineke from 2015, and appears as a consequence of the Decomposition Theorem applied to the forgetful map from a parabolic moduli space. We give a detailed description of the topology of this map, and that of the relevant local systems. Our work is self-contained, geometric and focuses on using the multiplicative structures of the theory.