Geometry of the intersection ring and vanishing relations in the cohomology of the moduli space of parabolic bundles on a curve

TL;DR

This paper investigates the structure of the cohomology ring of the moduli space of parabolic bundles on a curve, revealing vanishing relations and providing geometric representatives for Chern classes.

Contribution

It introduces a geometric approach to describe Chern classes and proves vanishing relations in the cohomology ring of the moduli space of parabolic bundles.

Findings

Chern classes have simple geometric representatives

The cohomology ring vanishes below the moduli space dimension

Supports analogy with the Newstead-Ramanan conjecture

Abstract

We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincar\'e duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern classes vanishes below the dimension of the moduli space, in analogy with the Newstead-Ramanan conjecture for stable bundles.