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Journal Article

Hilbert squares: derived categories and deformations


Belmans,  Pieter
Max Planck Institute for Mathematics, Max Planck Society;

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Belmans, P., Fu, L., & Raedschelders, T. (2019). Hilbert squares: derived categories and deformations. Selecta Mathematica, 25(3): 37. doi:10.1007/s00029-019-0482-y.

Cite as: https://hdl.handle.net/21.11116/0000-0004-6AB4-A
For a smooth projective variety $X$ with exceptional structure sheaf, and
$\operatorname{Hilb}^2X$ the Hilbert scheme of two points on $X$, we show that the Fourier-Mukai functor $\mathbf{D}^{\mathrm{b}}(X)
\to\mathbf{D}^{\mathrm{b}}(\operatorname{Hilb}^2X)$ induced by the universal ideal sheaf is fully faithful, provided the dimension of $X$ is at least 2. This fully faithfulness allows us to construct a spectral sequence relating the deformation theories of $X$ and $\operatorname{Hilb}^2X$ and to show that it degenerates at the second page, giving a Hochschild-Kostant-Rosenberg-type
filtration on the Hochschild cohomology of $X$. These results generalise known results for surfaces due to Krug-Sosna, Fantechi and Hitchin. Finally, as a by-product, we discover the following surprising phenomenon: for a smooth projective variety of dimension at least 3 with exceptional structure sheaf, it is rigid if and only if its Hilbert scheme of two points is rigid. This last fact contrasts drastically to the surface case: non-commutative deformations of a surface contribute to commutative deformations of its Hilbert square.