Variation of geometric invariant theory quotients and derived categories

Matthew Ballard, David Favero, Ludmil Katzarkov

Research output: Contribution to journalArticle

8 Citations (Scopus)

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 we completely describe all components appearing in these semi-orthogonal decompositions. We show how this general framework encompasses many well-known semi-orthogonal decompositions. We then proceed to give applications of this complete description. In this setting, we 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 and extend Orlov's σ-model/Landau-Ginzburg model correspondence.

Original languageEnglish (US)
Pages (from-to)235-303
Number of pages69
JournalJournal fur die Reine und Angewandte Mathematik
Volume2019
Issue number746
DOIs
StatePublished - Jan 1 2019

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Geometric Invariant Theory
Derived Category
Orthogonal Decomposition
Ginzburg-Landau Model
Quotient
Decomposition
Equivalence
Coherent Sheaf
Rational Curves
Factorization
Linearization
Moduli Space
Correspondence
Verify
Theorem
Model

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Variation of geometric invariant theory quotients and derived categories. / Ballard, Matthew; Favero, David; Katzarkov, Ludmil.

In: Journal fur die Reine und Angewandte Mathematik, Vol. 2019, No. 746, 01.01.2019, p. 235-303.

Research output: Contribution to journalArticle

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