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Variation of geometric invariant theory quotients and derived categories

  • Matthew Ballard EMAIL logo , David Favero and Ludmil Katzarkov

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.

Award Identifier / Grant number: DMS 0636606 RTG

Award Identifier / Grant number: DMS 0838210 RTG

Award Identifier / Grant number: DMS 0854977 FRG

Award Identifier / Grant number: DMS 0600800

Award Identifier / Grant number: DMS 0652633 FRG

Award Identifier / Grant number: DMS 0854977

Award Identifier / Grant number: MS 0901330

Funding source: Austrian Science Fund

Award Identifier / Grant number: P24572 N25

Award Identifier / Grant number: P20778

Funding statement: The first named author was funded by NSF DMS 0636606 RTG, NSF DMS 0838210 RTG, and NSF DMS 0854977 FRG. The second and third named authors were funded by NSF DMS 0854977 FRG, NSF DMS 0600800, NSF DMS 0652633 FRG, NSF DMS 0854977, NSF DMS 0901330, FWF P24572 N25, by FWF P20778 and by an ERC Grant.

Acknowledgements

The authors have benefited immensely from conversations and correspondence with Yujiro Kawamata, Manfred Herbst, Colin Diemer, Gabriel Kerr, Alastair Craw, Sukhendu Mehrotra, Andrei Căldăraru, Paolo Stellari, Ed Segal, Michael Thaddeus, Dmitri Orlov, Alexei Bondal, Kentaro Hori, Will Donovan, R. Paul Horja, Dragos Deliu, M. Umut Isik, Pawel Sosna, Emanuele Macrì, Alexander Kuznetsov, Daniel Halpern-Leistner, and Maxim Kontsevich and would like to thank them all for their time, patience, and insight.

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Received: 2015-09-08
Published Online: 2016-02-13
Published in Print: 2019-01-01

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