Discriminants and semi-orthogonal decompositions

TL;DR

This paper explores semi-orthogonal decompositions of derived categories of toric varieties, establishing a Jordan-Holder property and connecting wall-crossing phenomena with mirror symmetry and monodromy.

Contribution

It proves the Jordan-Holder property for these decompositions and formulates and verifies a conjecture linking discriminant intersection multiplicities with semi-orthogonal decomposition multiplicities.

Findings

Decompositions are independent of wall-crossing choices.

Wall-crossing yields derived autoequivalences for Calabi-Yau toric varieties.

Conjecture relating discriminant multiplicities to semi-orthogonal decomposition multiplicities is proven in some cases.

Abstract

The derived categories of toric varieties admit semi-orthogonal decompositions coming from wall-crossing in GIT. We prove that these decompositions satisfy a Jordan-Holder property: the subcategories that appear, and their multiplicities, are independent of the choices made. For Calabi-Yau toric varieties wall-crossing instead gives derived equivalences and autoequivalences, and mirror symmetry relates these to monodromy around the GKZ discriminant locus. We formulate a conjecture equating intersection multiplicities in the discriminant with the multiplicities appearing in certain semi-orthogonal decompositions. We then prove this conjecture in some cases.