New derived symmetries of some hyperk\"ahler varieties

TL;DR

This paper introduces new autoequivalences of derived categories for certain hyperk"ahler varieties, expanding the understanding of their symmetries through novel functor constructions.

Contribution

It constructs new autoequivalences for Hilbert schemes of points on K3 surfaces and cubic 4-folds using spherical and P-functors, proposing a generalization to all moduli spaces of sheaves.

Findings

Constructed autoequivalence for Hilb^2 and cubic 4-fold varieties.

Developed a theory of P-functors for n > 2.

Proposed conjecture for autoequivalences on all moduli spaces of sheaves.

Abstract

We construct a new autoequivalence of the derived category of the Hilbert scheme of n points on a K3 surface, and of the variety of lines on a smooth cubic 4-fold. For Hilb^2 and the variety of lines, we use the theory of spherical functors; to deal with Hilb^n for n > 2 we develop a theory of P-functors. We conjecture that the same construction yields an autoequivalence for any moduli space of sheaves on a K3 surface. In an appendix we give a cohomology and base change criterion which is well-known to experts, but not well-documented.