On the cohomology ring of the moduli space of rank 2 vector bundles on a curve

TL;DR

This paper investigates the rational cohomology ring of the moduli space of rank 2 vector bundles with fixed determinant over a curve, establishing relations, invariants, and a basis, confirming Mumford's conjecture.

Contribution

It proves properties of the cohomology ring, including a recurrence relation, a complete intersection structure for invariants, and verifies Mumford's structural formula.

Findings

First relation satisfies a recurrence in genus g

Invariant subring is a complete intersection ring

A Gr"obner basis for invariant relations is found

Abstract

Let $\MS_g$ be the moduli space of stable holomorphic vector bundles of rank 2 and fixed determinant of odd degree over a smooth complex projective curve of genus $g$. This paper proves various properties of the rational cohomology ring $H^*(\MS_g)$. It is shown that the first relation in genus $g$ between the standard generators satisfies a recurrence relation in $g$ and that the invariant subring for the mapping class group is a complete intersection ring. (These two results have been obtained independently by Zagier, Baranovsky and Siebert & Tian.) A Gr\"obner basis is found for the ideal of invariant relations. A structural formula for $H^*(\MS_g)$ (originally conjectured by Mumford) is verified and a natural monomial basis is given.