Universal functors on symmetric quotient stacks of Abelian varieties

Abstract

We consider certain universal functors on symmetric quotient stacks of Abelian varieties. In dimension two, we discover a family of $\mathbb{P}$-functors which induce new derived autoequivalences of Hilbert schemes of points on Abelian surfaces; a set of braid relations on a holomorphic symplectic sixfold; and a pair of spherical functors on the Hilbert square of an Abelian surface, whose twists are related to the well-known Horja twist. In dimension one, our universal functors are fully faithful, giving rise to a semiorthogonal decomposition for the symmetric quotient stack of an elliptic curve (which we compare to the one discovered by Polishchuk--Van den Bergh), and they lift to spherical functors on the canonical cover, inducing twists which descend to give new derived autoequivalences here as well.

Minor revisions. 35 pages

References in corpus (8)

• Calabi-Yau and fractional Calabi-Yau categories
• Moduli spaces of torsion sheaves on K3 surfaces and derived equivalences
• Equivalences of equivariant derived categories
• Integral Transforms and Deformations of K3 Surfaces
• Mukai flops and P-twists
• P-functor versions of the Nakajima operators
• A note on spherical functors
• Varieties with $\mathbb{P}$-units

Cited by in corpus (3)

• ${\mathbb{P}}^n$-functors
• A Semi-orthogonal Sequence in the Derived Category of the Hilbert Scheme of Three Points
• Hochschild cohomology and deformations of $\mathbb{P}$-functors