Variation of geometric invariant theory quotients and derived categories

TL;DR

This paper explores how derived categories of gauged Landau-Ginzburg models change under GIT variations, establishing semi-orthogonal decompositions and confirming the equivalence of D- and K- equivalences, with applications to toric stacks and moduli spaces.

Contribution

It provides a new framework linking GIT variations to derived categories, confirming D- and K- equivalence coincide, and applies these results to toric stacks and moduli spaces.

Findings

Derived categories are comparable via semi-orthogonal decompositions.

D- and K- equivalences coincide for the studied variations.

All projective toric Deligne-Mumford stacks have full exceptional collections.

Abstract

We study the relationship between derived categories of factorizations on gauged Landau-Ginzburg models related by variations of the linearization in Geometric Invariant Theory. Under assumptions on the variation, we show the derived categories are comparable by semi-orthogonal decompositions and describe the complementary components. We also verify a question posed by Kawamata: we show that $D$-equivalence and $K$-equivalence coincide for such variations. The results are applied to obtain a simple inductive description of derived categories of coherent sheaves on projective toric Deligne-Mumford stacks. This recovers Kawamata's theorem that all projective toric Deligne-Mumford stacks have full exceptional collections. Using similar methods, we prove that the Hassett moduli spaces of stable symmetrically-weighted rational curves also possess full exceptional collections. As a final application, we show how our results recover Orlov's $\sigma$-model/Landau-Ginzburg model correspondence.