Moduli spaces of rank 3 parabolic bundles over a many-punctured surface

Abstract

Let $M$ be the moduli space of rank 3 parabolic vector bundles over a Riemann surface with several punctures. By the Mehta-Seshadri correspondence, this is the space of rank 3 unitary representations of the fundamental group of the punctured surface with specified conjugacy classes of the images of each boundary component, up to conjugation by elements of the unitary group. For each puncture we consider the torus bundle on $M$ consisting of those representations where the image of the corresponding boundary component is a fixed element of the torus. We associate line bundles to these torus bundles via one-dimensional torus representations, and consider the subring of the cohomology ring of $M$ generated by their first Chern classes. By finding explicit sections of these line bundles with no common zeros we give a geometric proof that particular products of these Chern classes vanish. We prove that the ring generated by these Chern classes vanishes below the dimension of the moduli space, in a generalisation of a conjecture of Newstead.