The Pontryagin Rings of Moduli Spaces of Arbitrary Rank Holomorphic Bundles Over a Riemann Surface

Abstract

The cohomology of M(n, d), the moduli space of stable holomorphic bundles of coprime rank n and degree d and fixed determinant, over a Riemann surface Σ of genus g ⩾ 2, has been widely studied from a wide range of approaches. Narasimhan and Seshadri [17] originally showed that the topology of M(n, d) depends only on the genus g rather than the complex structure of Σ. An inductive method to determine the Betti numbers of M(n, d) was first given by Harder and Narasimhan [7] and subsequently by Atiyah and Bott [1]. The integral cohomology of M(n, d) is known to have no torsion [1] and a set of generators was found by Newstead [19] for n = 2, and by Atiyah and Bott [1] for arbitrary n. Much progress has been made recently in determining the relations that hold amongst these generators, particularly in the rank two, odd degree case which is now largely understood. A set of relations due to Mumford in the rational cohomology ring of M(2, 1) is now known to be complete [14]; recently several authors have found a minimal complete set of relations for the ‘invariant’ subring of the rational cohomology of M(2, 1) [2, 13, 20, 25]. Unless otherwise stated all cohomology in this paper will have rational coefficients.

Locations

• Journal of the London Mathematical Society - View
• arXiv (Cornell University) - PDF