THE INTEGRAL COHOMOLOGY OF THE HILBERT SCHEME OF TWO POINTS

Burt Totaro

doi:10.1017/fms.2016.5

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.

Highlights

In one way, things are unexpectedly good: the Hilbert scheme X [2] has torsionfree cohomology if X does (Theorem 2.2)
To explain one difficulty: some cohomology classes on X [2] can be defined as the classes c The Author 2016
Because the Hilbert scheme is only defined for complex manifolds, it is harder to construct ‘interesting’ classes on X [2] associated to arbitrary cohomology classes on X, for example to odd-degree cohomology classes

Summary

Main results

As in the proof of Lemma 3.1, the boundary in H 2r (EX , F2) of the product b[S2 Z − Z ] is the pushforward of the cohomology class b on PR(T ∗ Z ) via the proper map PR(T ∗ Z ) → PC(T ∗ X ). The elements g∗(zi ⊗z j ) in the cohomology of S2 X − X are the restrictions of the Borel–Moore homology classes f∗(zi ⊗ z j ) on S2 X , where f : X × X → S2 X is the obvious map These classes in H∗BM S2 X are linearly independent for i < j by Theorem 4.1.

Torsion-free cohomology

Complex submanifolds