The cohomology rings of moduli spaces of bundles over Riemann surfaces

Abstract

The cohomology of the moduli space JI(n, d) of semistable bundles of coprime rank n and degree d over a fixed Riemann surface M of genus g > 2 has been much studied over the last two decades, and a great deal is known about it. It is known that the cohomology has no torsion [2], the Betti numbers have been computed using various different methods [2, 5, 11], and a set of generators for the cohomology ring is known [2]. The purpose of this article is to provide a method of deciding when an expression in these generators is a relation, and in particular, when the rank n is two, to prove two conjectures about the relations between the generators due to Mumford and to Newstead and Ramanan. Newstead and Ramanan conjectured that certain relations hold among the generators, whereas Mumford conjectured that a certain set of relations is a complete set. In fact the conjecture of Newstead and Ramanan has two parts, the first of which was proved by Gieseker in 1982 [4]. The second part of their conjecture, which is the part examined in this article, was given an independent proof recently by Thaddeus [19], who has a very different method of obtaining information about the relations in the cohomology ring. The computation of the Betti numbers of the moduli spaces .I'(n, d) in [2] stimulated a general method of computing Betti numbers of quotients in the sense of Mumford's geometric invariant theory [14] of nonsingular complex projective varieties by complex reductive group actions (see [8, 9, 10]). Similarly the ideas applied in this article to finding generators and relations for the cohomology rings of the moduli spaces .I'(n, d) also lead to an analogous method of obtaining generators and relations for the rational cohomology rings of geometric invariant theoretic quotients. The first section provides background material on the cohomology of the moduli spaces .I'(n, d) . In the second section, Mumford's conjecture is proved when the rank n is two. The method could, in principle, be used to decide whether the conjecture is true for rank n > 2 but appears to become significantly more complicated in the general case. The third section gives a method for deciding when an expression in the generators is a relation, and the fourth