Variation of geometric invariant theory quotients and derived categories

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 $σ$-model/Landau-Ginzburg model correspondence.

Theorems & Definitions (225)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 2.1.1
• Remark 2.1.2
• Definition 2.1.3
• Definition 2.1.4
• Definition 2.1.5
• Remark 2.1.6