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. 1The physicists, M. Herbst, K. Hori, and D. Page, studied Abelian gauged linear σ-models in [HHP08], where they rediscovered windows and used it to explain Orlov's theorem. Through work of E. Witten [Wit93], the phases of gauged linear σ-model are exactly the different chambers of the GIT fan for the action. So VGIT is an implicit piece of [HHP08]. Building on the ideas of [HHP08], E. Segal re-proved the Calabi-Yau hypersurface case of Orlov's theorem using VGIT, LG-models, and windows. Segal's development and presentation of the ideas left an indelible mark on the authors of this paper. Subsequently, Shipman extended Segal's methods to handle the Calabi-Yau complete intersection case of Orlov's theorem [Shi10]. Independently, Herbst and J. Walcher extended Orlov's theorem to Calabi-Yau complete intersections in toric varieties [HW12] and used it to study auto-equivalences. Along this vein, W. Donovan constructed exotic derived equivalences via Grassmannian twists [Don11]. Donovan's work represents the first application of these ideas outside the Abelian realm.Two additional papers on related material appeared contemporaneously to this paper. Both are independent. The first is due to D. Halpern-Leistner [H-L12] and has significant overlap with this paper. Halpern-Leistner proves the existence of the fully-faithful functors, Φ + d and Φ − d , and equivalences for µ = 0 in Theorem 1 when G is not necessarily Abelian. He also gives a definition of windows in the non-Abelian setting. The authors and Halpern-Leistner interacted during a conference at the University of Miami where the first and second authors presented preliminary results of this paper. Halpern-Leistner informed the authors of his work and later provided a preprint version of [H-L12] while the first version of this paper was in preparation. The second paper is due to Donovan and Segal [DS12] and builds on [Don11]. Again studying Grassmannian twists, Donovan and Segal use a different definition of window built from M. Kapranov's exceptional collection [Kap88]. Neither of the concurr...

Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks

“…Instead, we expect that the quantum cohomology of X + would contain the quantum cohomology of X − as a direct summand after analytic continuation. This is analogous to the conjecture that D b (X + ) contains D b (X − ) as a semiorthogonal summand [13,71,72,9], where D b (X ± ) denotes the derived category of coherent sheaves on X ± . In this paper, we describe a decomposition of quantum cohomology D-modules (and of all genus Gromov-Witten potentials) for toric Deligne-Mumford stacks under discrepant transformations.…”

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On derived categories of arithmetic toric varieties

“…The endpoint of this run is P n , and the birational map P n V n is described above (blow up the torus invariants points of P n and then inductively flip the linear subspaces of dimension d < m, where n = 2m). To understand how the derived category is affected under such modifications, it will be advantageous to present the process as a variation of GIT quotients of the spectrum of the Cox ring of the blow up of P n using [BFK17,Bal17].…”

Perverse schobers on Riemann surfaces: constructions and examples

“…I review results from GIT sufficient to recall, from previous work [12], a construction of a spherical pair for certain GIT wall crossings: this is Theorem 5.9. For further details on GIT, I refer the reader to treatments of Halpern-Leistner [20], and Ballard, Favero, and Katzarkov [4].…”

“…The T w are spherical twists obtained by Halpern-Leistner and Shipman [21], and the Φ w are window equivalences as given by Halpern-Leistner [20], and Ballard-Favero-Katzarkov [4].…”