Hilbert squares: derived categories and deformations

Abstract

For a smooth projective variety X with exceptional structure sheaf, and X [2] the Hilbert scheme of two points on X , we show that the Fourier-Mukai functor D b (X ) → D b (X [2] ) 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 X [2] 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 byproduct, 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.

Derived categories of (nested) Hilbert schemes

“…For n ≤ − 1 the derived category is expected to be indecomposable. So (5) would be a decomposition into indecomposable pieces. For the proof of theorem D and more context, see section 6.…”

“…In the past two decades their derived categories have been studied thoroughly, especially in the context of the derived McKay correspondence [10,16]. They provide an important source of interesting behaviour for derived categories of smooth projective varieties, such as the construction of non-standard autoequivalences [1,20,22] or interesting fully faithful functors [24,5]. A good understanding of their derived categories, and functors relating them, has e.g.…”

“…A good understanding of their derived categories, and functors relating them, has e.g. led to a more abstract interpretation of their deformation theory [5].…”

The Fourier-Mukai transform of a universal family of stable vector bundles

“…For example Krug and Sosna prove in [10] that the Fourier-Mukai transform Φ I Z : D b (S) → D b (S [n] ) induced by the universal ideal sheaf I Z of the Hilbert scheme S [n] is fully faithful for a surface S with p g = q = 0, hence D b (S) is an admissible subcategory in D b (S [n] ). This result was generalized for the Hilbert square X [2] to smooth projective varieties X with exceptional structure sheaf and arbitrary dimension dim(X) ≥ 2, see [1].…”

Smooth components on special iterated Hilbert schemes

“…Acknowledgement. I thank Pieter Belmans for informing me about the fully faithfulness results in [3] and [11] as well as Ziyu Zhang for many useful conversations.…”