Theta Functions and Hodge Numbers of Moduli Spaces of Sheaves on Rational Surfaces

Abstract:

We compute generating functions for the Hodge numbers of the moduli spaces of H-stable rank 2 sheaves on a rational surface S in terms of theta functions for indefinite lattices. If H lies in the closure of the ample cone and has self-intersection 0, it follows that the generating functions are Jacobi forms. In particular the generating functions for the Euler numbers can be expressed in terms of modular forms, and their transformation behaviour is compatible with the predictions of S-duality. We also express the generating functions for the signatures in terms of modular forms. It turns out that these generating functions are also (with respect to another developing parameter) the generating function for the Donaldson invariants of S evaluated on all powers of the point class.