The Mumford relations and the moduli of rank three stable bundles

Abstract

The cohomology ring of the moduli space $M(n,d)$ of semistable bundles of coprime rank $n$ and degree $d$ over a Riemann surface $M$of genus $g \geqslant 2$ has again proven a rich source of interest in recent years. The rank two, odd degree case is now largely understood. In 1991 Kirwan [8] proved two long standing conjectures due to Mumford and to Newstead and Ramanan. Mumford conjectured that a certain set of relations form a complete set; the Newstead-Ramanan conjecture involved the vanishing of the Pontryagin ring. The Newstead–Ramanan conjecture was independently proven by Thaddeus [15] as a corollary to determining the intersection pairings. As yet though, little work has been done on the cohomology ring in higher rank cases. A simple numerical calculation shows that the Mumford relations themselves are not generally complete when $n>2$. However by generalising the methods of [8] and by introducing new relations, in a sense dual to the original relations conjectured by Mumford, we prove results corresponding to the Mumford and Newstead-Ramanan conjectures in the rank three case. Namely we show (Sect. 4) that the Mumford relations and these ‘dual’ Mumford relations form a complete set for the rational cohomology ring of $M(3,d)$ and show (Sect. 5) that the Pontryagin ring vanishes in degree$12g-8$ and above.