The integral cohomology of the Hilbert scheme of points on a surface

TL;DR

This paper proves that the Hilbert scheme of points on a smooth complex projective surface with torsion-free cohomology also has torsion-free cohomology, extending previous results to a broader class of surfaces.

Contribution

It establishes torsion-freeness of cohomology for Hilbert schemes on a wider class of surfaces using reduced obstruction theory.

Findings

Hilbert schemes of points on certain surfaces have torsion-free cohomology

Extension of Markman's results to more general surfaces

Application of reduced obstruction theory to nested Hilbert schemes

Abstract

We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme X^[n] has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas's reduced obstruction theory for nested Hilbert schemes of surfaces.