Universal functors on symmetric quotient stacks of Abelian varieties

TL;DR

This paper introduces universal functors on symmetric quotient stacks of Abelian varieties, revealing new autoequivalences, braid relations, and semiorthogonal decompositions in various dimensions, advancing the understanding of derived categories in algebraic geometry.

Contribution

It uncovers new $bP$-functors and spherical functors on Abelian varieties' symmetric stacks, leading to novel autoequivalences and relations, extending prior work in derived category theory.

Findings

Discovery of $bP$-functors inducing autoequivalences on Hilbert schemes

Identification of braid relations on a holomorphic symplectic sixfold

Semiorthogonal decomposition for symmetric quotient stacks of elliptic curves

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.