Discriminants and Semi-orthogonal Decompositions

Alex Kite,Ed Segal

doi:10.1007/s00220-021-04298-2

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–Hölder 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.

Highlights

Let X be a toric variety, constructed as a GIT quotient of a vector space V by a torusT
If we cross to a quotient X, and K X is ‘more negative’ than K X, Db(X ) decomposes as
Since the extra pieces are always equivalent to the derived category of a toric variety they themselves can be decomposed by the same procedure, and we get a recursive algorithm which terminates after a finite number of steps

Summary

Introduction

Let X be a toric variety, constructed as a GIT quotient of a vector space V by a torus. If we split V by weights as V+ ⊕ V0 ⊕ V− it’s easy to see that X± is a vector bundle over PV± × V0, where PV± is a weighted projective space In this rank 1 case the discriminant locus is always a single point δ so the FIPS is C∗ \δ (see Example 4.4). Aspinwall–Plesser–Wang [APW] observed that there is a correspondence between these components ∇i and certain toric varieties Zi , built from subsets of the original toric data They conjecture that for each phase X there should be a spherical functor. They only consider the case when Z is projective, meaning that the decomposition of Db(Z ) is a full exceptional collection, and they conjecture that the number of exceptional objects agrees with the intersection multiplicity of ∇ with CW [HLSh, Remark 4.7]. Our conjecture is a synthesis of theirs with the work of [APW]

Toric Background

Semi-orthogonal Decompositions for Toric Varieties

FI Parameter Spaces and Discriminants