A Note on the Cohomology of Moduli of Rank Two Stable Bundles

TL;DR

This paper revisits the cohomology of moduli spaces of rank two stable bundles, confirming that a specific set of relations derived from Mumford's conjecture generate the entire cohomology ring, using Kirwan's methods.

Contribution

It provides a new proof that the first three Mumford relations generate the relation ideal and extends Mumford's conjecture to show these relations generate the entire cohomology ring as a module.

Findings

First three Mumford relations generate the relation ideal.

Relations from the first vanishing Chern class generate the entire cohomology ring.

The results follow directly from Kirwan's original calculations.

Abstract

The rational cohomology of the moduli space of rank two, odd degree stable bundles over a curve (of genus g > 1) has been studied intensely in recent years and in particular the invariant subring generated by Newstead's generators alpha, beta, gamma. Several authors have independently found a minimal complete set of relations for this subring. Their methods are very different from the methods originally employed by Kirwan to prove Mumford's conjecture -- that relations derived from the vanishing Chern classes of a particular rank 2g-1 bundle are a complete set of relations for the entire cohomology ring. This note contains two theorems which readily follow from Kirwan's original calculations. We rederive the above result showing that the first three invariant Mumford relations generate the relation ideal of the invariant subring. Secondly we prove a stronger version of Mumford's conjecture and show that the relations coming from the first vanishing Chern class generate the relation ideal of the entire cohomology ring as a Q[alpha,beta]-module. (Only a few typos have been amended in this revised version).