Moduli of sheaves on surfaces and action of the oscillator algebra

Abstract

This paper gives a generalization of some results on Hilbert schemes of points on surfaces. Let MG(r, n) (resp. MU(r, n)) be the Gieseker (resp. Uhlenbeck) compactification of the moduli spaces of stable bundles on a smooth projective surface. We show that, for surfaces satisfying some technical condition: (a) The natural map MG(r, n) → MU(r, n) generalizing the Hilbert- Chow morphism from the Hilbert scheme of n points on S to the n-th symmetric power, is strictly semi-small in the sense of Goresky- MacPherson with respect to some stratification. (b) Let Pt(X) be the Intersection Homology Poincare polynomial of X. Generalizing the computation due to Gottsche and Sorgel we prove that the ratio Σn qnPt(MG (r, n)) / Σn qnPt(MU (r, n)) is a character of a certain Heisenberg-type algebra. (c) Generalizing results of Nakajima we show how to obtain the action of the Heisenberg algebra on the cohomology using correspondences. © 2000 Applied Probability Trust.