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Non-commutative methods in deformation theory of Hilbert schemes of points on surfaces.

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EMG - New equivariant methods in algebraic and differential geometry

Abstract: We study the deformation theory of Hilbert schemes of points on surfaces by looking more broadly at the deformation theory of their derived categories, which is controlled by the Hochschild cohomology. In this way, we recover, unify, and extend the previous works of Fantechi, Hitchin, and Boissière. One interesting finding is that the Hochschild cohomology of a Hilbert scheme of a surface not only depends on that of the surface, but also on the more generally bigraded cohomology theory called Hochschild-Serre cohomology of the surface. Our method computes the Hochschild-Serre cohomology of the symmetric stack [X^n/S_n] in terms of the Hochschild-Serre cohomology of X. This is based on a joint work with Pieter Belmans and Andreas Krug, arXiv:2309.06244.

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