Hochschild cohomology of Hilbert schemes of points on surfaces

TL;DR

This paper computes the Hochschild cohomology of Hilbert schemes of points on surfaces, revealing it depends on Hochschild-Serre cohomology, and explores implications for deformation theory.

Contribution

It introduces the concept of Hochschild-Serre cohomology and computes it for Hilbert schemes, extending understanding of their deformation properties.

Findings

Hochschild cohomology of Hilbert schemes depends on Hochschild-Serre cohomology.

Provides formulas relating Hochschild-Serre cohomology of Hilbert schemes to that of the surface.

Extends deformation theory results for Hilbert schemes.

Abstract

We compute the Hochschild cohomology of Hilbert schemes of points on surfaces and observe that it is, in general, not determined solely by the Hochschild cohomology of the surface, but by its "Hochschild-Serre cohomology": the bigraded vector space obtained by taking Hochschild homologies with coefficients in powers of the Serre functor. As applications, we obtain various consequences on the deformation theory of the Hilbert schemes; in particular, we recover and extend results of Fantechi, Boissi\`ere, and Hitchin. Our method is to compute more generally for any smooth proper algebraic variety $X$ the Hochschild-Serre cohomology of the symmetric quotient stack $[X^n/\mathfrak{S}_n]$, in terms of the Hochschild-Serre cohomology of $X$.