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

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.

Keywords

• 14C05
• 55R80
• Hilbert scheme of points
• integral cohomology
• Chow motive