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...

We provide a factorization model for the continuous internal Hom, in the homotopy category of k-linear dg-categories, between dg-categories of equivariant factorizations. This motivates a notion, similar to that of Kuznetsov, which we call the extended Hochschild cohomology algebra of the category of equivariant factorizations. In some cases of geometric interest, extended Hochschild cohomology contains Hochschild cohomology as a subalgebra and Hochschild homology as a homogeneous component. We use our factorization model for the internal Hom to calculate the extended Hochschild cohomology for equivariant factorizations on affine space.Combining the computation of extended Hochschild cohomology with the Hochschild-Kostant-Rosenberg isomorphism and a theorem of Orlov recovers and extends Griffiths' classical description of the primitive cohomology of a smooth, complex projective hypersurface in terms of homogeneous pieces of the Jacobian algebra. In the process, the primitive cohomology is identified with the fixed subspace of the cohomological endomorphism associated to an interesting endofunctor of the bounded derived category of coherent sheaves on the hypersurface. We also demonstrate how to understand the whole Jacobian algebra as morphisms between kernels of endofunctors of the derived category.Finally, we present a bootstrap method for producing algebraic cycles in categories of equivariant factorizations. As proof of concept, we show how this reproves the Hodge conjecture for all self-products of a particular K3 surface closely related to the Fermat cubic fourfold. BALLARD, FAVERO, AND KATZARKOV

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The Orlov spectrum is a new invariant of a triangulated category. It was introduced by D. Orlov building on work of A. Bondal-M. van den Bergh and R. Rouquier. The supremum of the Orlov spectrum of a triangulated category is called the ultimate dimension. In this work, we study Orlov spectra of triangulated categories arising in mirror symmetry. We introduce the notion of gaps and outline their geometric significance. We provide the first large class of examples where the ultimate dimension is finite: categories of singularities associated to isolated hypersurface singularities. Similarly, given any nonzero object in the bounded derived category of coherent sheaves on a smooth Calabi-Yau hypersurface, we produce a new generator by closing the object under a certain monodromy action and uniformly bound this new generator's generation time. In addition, we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived Fukaya category of a symplectic surface of genus greater than one.Comment: Previous version was missing its head, 52 pages, 1 figure, uses Tikz; comments are still encouraged

Abstract. Building upon ideas of Eisenbud, Buchweitz, Positselski, and others, we introduce the notion of a factorization category. We then develop some essential tools for working with factorization categories, including constructions of resolutions of factorizations from resolutions of their components and derived functors. Using these resolutions, we lift fully-faithfulness and equivalence statements from derived categories of Abelian categories to derived categories of factorizations. Some immediate geometric consequences include a realization of the derived category of a projective hypersurface as matrix factorizations over a noncommutative algebra and a generalization of a theorem of Baranovsky and Pecharich.