On the cobordism class of the Hilbert scheme of a surface

Let S be a smooth projective surface and S-[n] the Hilbert scheme of zero-d imensional subschemes of S of length n. We prove that the class of S-[n] in the complex cobordism ring depends only on the class of the surface itself . Moreover, we compute the cohomology and holomorphic Euler characteristics of certain tautological sheaves on S-[n] and prove results on the general structure of certain integrals over polynomials in Chern classes of tautolo gical sheaves.