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.
Highlights
The derived category of coherent sheaves on a variety is a fundamental geometric invariant with fascinating and intricate connections to birational geometry, mirror symmetry, non-commutative geometry and representation theory, to name but a few
We can express A[n] as a moduli space of ideal sheaves on A equipped with a universal sheaf U = IZ on A × A[n] where Z ⊂ A × A[n] is the universal family of length n subschemes of A
We show that the restriction FA×{x} : D( A) → D(Kn−1) of this functor to any fibre over a point in the second factor coincides with the Pn−1-functor FK considered in [36, Theorem 4.1]; in particular, F is a family version of FK
Summary
The derived category of coherent sheaves on a variety is a fundamental geometric invariant with fascinating and intricate connections to birational geometry, mirror symmetry, non-commutative geometry and representation theory, to name but a few. It is fair to say that derived categories are ubiquitous in mathematics. Just as equivalences between derived categories of different varieties can indicate deep and important con-
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