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 language | English (US) |
|---|---|
| Pages (from-to) | 235-303 |
| Number of pages | 69 |
| Journal | Journal fur die Reine und Angewandte Mathematik |
| Volume | 2019 |
| Issue number | 746 |
| DOIs | |
| State | Published - Jan 1 2019 |
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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 journal › Article
}
TY - JOUR
T1 - Variation of geometric invariant theory quotients and derived categories
AU - Ballard, Matthew
AU - Favero, David
AU - Katzarkov, Ludmil
PY - 2019/1/1
Y1 - 2019/1/1
N2 - 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.
AB - 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.
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U2 - 10.1515/crelle-2015-0096
DO - 10.1515/crelle-2015-0096
M3 - Article
AN - SCOPUS:85057391509
VL - 2019
SP - 235
EP - 303
JO - Journal fur die Reine und Angewandte Mathematik
JF - Journal fur die Reine und Angewandte Mathematik
SN - 0075-4102
IS - 746
ER -