Complete sets of relations in the cohomology rings of moduli spaces of holomorphic bundles and parabolic bundles over a Riemann surface

The cohomology of the moduli space M(n, d) of stable holomorphic vector bundles of coprime rank n and degree d over a fixed compact Riemann surface Σ of genus g ≥ 2 has been intensely studied over many years. In the case when n = 2 we now have a very thorough understanding of its structure [3, 24, 49, 56]. For arbitrary n it is known that the cohomology has no torsion [1], and inductive and closed formulas for computing the Betti numbers have been obtained [1, 8, 9, 19, 32], as well as a set of generators for the cohomology ring [1]. When n = 2 the relations between these generators can be explicitly described [3, 20, 24, 49, 51, 56] and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is now known to be true [29, 56]. Less is known about the relations between the generators when n > 2, although in principle these can be obtained from the formulas given in [23, 55] for the evaluation of polynomials in the generators on the fundamental class of M(n, d). It was shown in [11] that the most obvious generalisation of Mumford’s conjecture to the cases when n > 2 is false, although a modified version of the conjecture (using ‘dual Mumford relations’ together with the original Mumford relations) is true for n = 3. In this paper we generalise the concept of the Mumford relations somewhat further and show that these generalised Mumford relations form a complete set for arbitrary rank n. The generators for H(M(n, d)) given by Atiyah and Bott in [1] are obtained from a (normalised) universal bundle V over M(n, d) × Σ. With respect to the Künneth decomposition of H(M(n, d)× Σ) the rth Chern class cr(V ) of V can be written as