Universal formulas for characteristic classes on the Hilbert schemes of points on surfaces

Abstract

This article can be seen as a sequel to the first author's article “Chern classes of the tangent bundle on the Hilbert scheme of points on the affine plane,” where he calculates the total Chern class of the Hilbert schemes of points on the affine plane by proving a result on the existence of certain universal formulas expressing characteristic classes on the Hilbert schemes in term of Nakajima's creation operators. The purpose of this work is (at least) two-fold. First of all, we clarify the notion of “universality” of certain formulas about the cohomology of the Hilbert schemes by defining a universal algebra of creation operators. This helps us to reformulate and extend a lot of the first author's previous results in a very precise manner. Secondly, we are able to extend the previously found results by showing how to calculate any characteristic class of the Hilbert scheme of points on the affine plane in terms of the creation operators. In particular, we have included the calculation of the total Segre class and the square root of the Todd class. Using this methods, we have also found a way to calculate any characteristic class of any tautological sheaf on the Hilbert scheme of points on the affine plane. This in fact gives another complete description of the ring structure of the cohomology spaces of the Hilbert schemes of points on the affine plane.