Hochschild cohomology of Hilbert schemes of points on surfaces

Research output: Working paper/Preprint › Preprint

Authors

  • Pieter Belmans
  • Lie Fu
  • Andreas Krug

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Original languageEnglish
Publication statusE-pub ahead of print - 12 Sept 2023

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 \(X\).

Keywords

    math.AG

Cite this

Hochschild cohomology of Hilbert schemes of points on surfaces. / Belmans, Pieter; Fu, Lie; Krug, Andreas.
2023.

Research output: Working paper/Preprint › Preprint

Belmans, P., Fu, L., & Krug, A. (2023). Hochschild cohomology of Hilbert schemes of points on surfaces. Advance online publication. https://doi.org/10.48550/arXiv.2309.06244
Belmans P, Fu L, Krug A. Hochschild cohomology of Hilbert schemes of points on surfaces. 2023 Sept 12. Epub 2023 Sept 12. doi: 10.48550/arXiv.2309.06244
Belmans, Pieter ; Fu, Lie ; Krug, Andreas. / Hochschild cohomology of Hilbert schemes of points on surfaces. 2023.
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