The integral cohomology of the Hilbert scheme of two points

TL;DR

This paper computes the mod 2 cohomology of the Hilbert scheme of two points on any complex manifold and the integral cohomology when the manifold has torsion-free cohomology, revealing subtle torsion phenomena.

Contribution

It provides the first comprehensive computation of the integral and mod 2 cohomology of X^{[2]} for complex manifolds, extending previous rational cohomology results.

Findings

Computed mod 2 cohomology of X^{[2]} for any complex manifold.

Determined integral cohomology of X^{[2]} when X has torsion-free cohomology.

Applications to the study of 2-torsion in Chow groups of cubic hypersurfaces.

Abstract

The Hilbert scheme X^{[a]} of points on a complex manifold X is a compactification of the configuration space of a-element subsets of X. The integral cohomology of X^{[a]} is more subtle than the rational cohomology. In this paper, we compute the mod 2 cohomology of X^{[2]} for any complex manifold X, and the integral cohomology of X^{[2]} when X has torsion-free cohomology. The results of this paper are used in Voisin's work on the universal CH_0 group of cubic hypersurfaces, because the crucial point there is to study the 2-torsion in the Chow group.