The First Two Betti Numbers of the Moduli Spaces of Vector Bundles on Surfaces

This paper is a continuation of our effort in understanding the geometry of the moduli space of stable vector bundles. For any polarized smooth projective surface (X,H) and for any choice of (I, d) ∈ Pic(X) × H(X,Z), there is a coarse moduli space M(I, d) of rank two μ-stable (with respect to H) locally free sheaves E of ∧E ∼= I and c2(E) = d. This moduli space has been studied extensively recently. One important discovery is that the moduli space M(I, d) exhibits remarkable properties at stable range. To cite a few, for arbitrary surface the moduli space M(I, d) has the expected dimension, is smooth at general points and is irreducible, and for a large class of surfaces of general type M(I, d) are of general type, all true for d sufficiently large [Fr, GL, Li2, Do, Zh]. In this paper, we will investigate another aspect of this moduli space. Namely, the Betti numbers of M(I, d). So far, there have been a lot of progress along this direction based on two different approaches: Algebro-geometric approach and gauge theoretic approach. The algebraic geometry approach is relatively new. In [ES,Ki,Yo], they studied in detail the Betti numbers of the moduli space of stable sheaves over P (for the rank two and higher rank cases). Beauville [Be] has a nice observation concerning some rational surfaces and Göttsche and Huybrechts [GH] have worked out the case for K3 surfaces. The gauge theory approach has been around for quite a while. To begin with, let (M,g) be a compact oriented Riemannian fourmanifold and let Pd be a smooth SO(3) (or SU(2))-bundle over M associated to a rank two vector bundle of c1 = I and c2 = d. Consider the pair