Moduli of sheaves on surfaces and action of the oscillator algebra

This thesis studies two compactifications of the moduli space of rank r vector bundles on a smooth complex projective surface S: the Gieseker compactification $MGr,n$ and the Uhlenbeck compactification $MUr,n$. The following results are proved for a surface S satisfying some technical condition: (a) The natural map $MGr,n\to MUr,n$ 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. (b) Let $PtX$ be the Intersection Homology Poincare polynomial of X. Generalizing the computation due to Göttsche and Sorgel, it is proved that the ratio SnqⁿPt&parl0; M^Gr,n&parr0; SnqⁿPt&parl0;M ^Ur,n&parr0; is a character of a certain Heisenberg-type algebra. (c) In generalization of results on Hilbert schemes of points due to Nakajima and Grojnowski, it is shown how to obtain the action of the Heisenberg algebra on the cohomology using correspondences. (d) As a by-product of the studies of the two compactifications, the thesis proves the irreducibility of the space of commuting nilpotent matrices.