Exceptional sequences and derived autoequivalences

TL;DR

This paper establishes a general theorem relating derived autoequivalences of a variety to the existence of a suitable functor from another variety with a full exceptional sequence, with applications to Calabi-Yau hypersurfaces.

Contribution

It introduces a new theorem linking derived autoequivalences to exceptional sequences, extending previous results to more general settings.

Findings

Derived autoequivalence relations are established using exceptional sequences.

The theorem applies to Calabi-Yau varieties and hypersurfaces in related varieties.

A resolution of the diagonal is constructed from the exceptional sequence.

Abstract

We prove a general theorem that gives a non trivial relation in the group of derived autoequivalences of a variety (or stack) X, under the assumption that there exists a suitable functor from the derived category of another variety Y admitting a full exceptional sequence. Applications include the case in which X is Calabi-Yau and either X is a hypersurface in Y (this extends a previous result by the author and R.L. Karp, where Y was a weighted projective space) or Y is a hypersurface in X. The proof uses a resolution of the diagonal of Y constructed from the exceptional sequence.