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

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.