New derived autoequivalences of Hilbert schemes and generalised Kummer varieties

Abstract

We show that for every smooth projective surface X and every $n\ge 2$ the push-forward along the diagonal embedding gives a $P^{n-1}$-functor into the $S_n$-equivariant derived category of X^n. Using the Bridgeland--King--Reid--Haiman equivalence this yields a new autoequivalence of the derived category of the Hilbert scheme of n points on X. In the case that the canonical bundle of X is trivial and n=2 this autoequivalence coincides with the known EZ-spherical twist induced by the boundary of the Hilbert scheme. We also generalise the 16 spherical objects on the Kummer surface given by the exceptional curves to n^4 orthogonal $P^{n-1}$-Objects on the generalised Kummer variety.

Mistakes in Lemma 3.3, Prop. 4.4, and Remark 4.7 corrected. The changes do not affect the main results

References in corpus (3)

• New derived symmetries of some hyperkähler varieties
• Derived autoequivalences of generalised Kummer varieties
• Tensor products of tautological bundles under the Bridgeland-King-Reid-Haiman equivalence

Cited by in corpus (6)

• Derived autoequivalences of generalised Kummer varieties
• Mukai flops and P-twists
• Formality of $\mathbb{P}$-objects
• ${\mathbb{P}}^n$-functors
• Spherical functors on the Kummer surface
• On the derived category of the Hilbert scheme of points on an Enriques surface