On the Cobordism Class of the Hilbert Scheme of a Surface

TL;DR

This paper demonstrates that the cobordism class of the Hilbert scheme of points on a surface depends solely on the surface's class, and it computes related cohomological invariants and integrals.

Contribution

It establishes the cobordism invariance of Hilbert schemes of surfaces and provides explicit calculations of cohomological and Euler characteristics.

Findings

Cobordism class of S^[n] depends only on S

Computed cohomology and Euler characteristics of tautological sheaves

Results on integrals over Chern classes of tautological sheaves

Abstract

Let S be a smooth projective surfaces and S^[n] the Hilbert scheme of zero-dimensional subschemes of S of length n. We proof 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 characterisitcs of certain tautological sheaves on S^[n] and prove results on the general structure of certain integrals over polynomials in Chern classes of tautological sheaves.