Probing Moduli Spaces of Sheaves with Donaldson and Seiberg Witten Invariants

Abstract

We use Donaldson invariants of regular surfaces with p_g >0 to make quantitative statements about modulispaces of stable rank 2 sheaves. We give two examples: a quantitative existence theorem for stable bundles, and a computation of the rank of the canonical holomorphic two forms on the moduli space. The results are in some sense dual to the Donaldson and O'Grady non vanishing theorems because they use the Donaldson series of the surface as _input_. Results in purely algebraic geometric terms can be obtained by using the explicit form of the Donaldson series of the surface. The Donaldson series are easy to compute using the Seiberg Witten invariants and the Witten conjecture which has recently been rigorously proved by Feehan and Leness.