1997/07/17 Warwick Symposium on Complex Differential Geometry, ``Homology of quot-schemes, blowup formulas and Fock spaces"

This is a preliminary report on my joint work with I.Grojnowski and K.Yoshioka.

Grojnowski and I independently showed that homoglogy groups of Hilbert schemes of points on a complex surface is a Fock space, i.e., the fundamental representation of the (infinite dimensional) Heisenberg algebra.

On the other hand, Yoshioka computed Betti numbers of the quot-schemes and effects of the blowups on homology groups of moduli spaces of torsion-free sheaves.

Using framed moduli spaces, we show that Yoshioka's formulas can be understood from our knowledge of homology groups of Hilbert schemes via the localization theorem.

The Fock space is a ``building block" in the representation theory of the affine Lie algebras. (e.g., various realizations of representations of affine Lie algebras can be constructed from Fock spaces.) We expect that Hilbert schemes will play a similar role in the moduli space of torsion-free sheaves.