Abstract. We prove a Knörrer periodicity type equivalence between derived factorization categories of gauged LG models, which is an analogy of a theorem proved by Shipman and Isik independently. As an application, we obtain a gauged LG version of Orlov's theorem describing a relationship between categories of graded matrix factorizations and derived categories of hypersurfaces in projective spaces, by combining the above Knörrer periodicity type equivalence and the theory of variations of GIT quotients due to Ballard, Favero and Katzarkov.
Abstract. For a given Fourier-Mukai equivalence of bounded derived categories of coherent sheaves on smooth quasi-projective varieties, we construct Fourier-Mukai equivalences of derived factorization categories of gauged Landau-Ginzburg (LG) models.As an application, we obtain some equivalences of derived factorization categories of K-equivalent gauged LG models. This result is an equivariant version of the result of Baranovsky and Pecharich, and it also gives a partial answer to Segal's conjecture. As another application, we prove that if the kernel of the Fourier-Mukai equivalence is linearizable with respect to a reductive affine algebraic group action, then the derived categories of equivariant coherent sheaves on the varieties are equivalent. This result is shown by Ploog for finite groups case.
We show that if X is a smooth quasi-projective 3-fold admitting a flopping contraction, then the fundamental group of an associated simplicial hyperplane arrangement acts faithfully on the derived category of X. The main technical advance is to use torsion pairs as an efficient mechanism to track various objects under iterations of the flop functor (respectively, mutation functor). This allows us to relate compositions of the flop functor (respectively, mutation functor) to the theory of Deligne normal form, and to give a criterion for when a finite composition of 3-fold flops can be understood as a tilt at a single torsion pair. We also use this technique to give a simplified proof of Brav-Thomas [BT] for Kleinian singularities.
For a separated Noetherian scheme X with an ample family of line bundles and a non-zero-divisor W ∈ Γ(X, L) of a line bundle L on X, we classify certain thick subcategories of the derived matrix factorization category DMF(X, L, W ) of the Landau-Ginzburg model (X, L, W ). Furthermore, by using the classification result and the theory of Balmer's tensor triangular geometry, we show that the spectrum of the tensor triangulated category (DMF(X, L, W ), ⊗ 1 2 ) is homeomorphic to the relative singular locus Sing(X 0 /X), introduced in this paper, of the zero scheme X 0 ⊂ X of W .1.2. Relative singular locus. To state our main results, we introduce a new notion of relative singular locus. Let i : T ֒→ S be a closed immersion of Noetherian schemes. We define the relative singular locus, denoted by SingT is worse than the mildness of the singularity of p in S. In fact, for a quasi-projective variety X over C and a regular function f ∈ Γ(X, O X ) which is non-zero-divisor, we have the following equality of subsets of the associated complex analytic space (X an , O an X ); Sing loc (f −1 (0)/X) ∩ X an = Crit(f an ) ∩ Zero(f an ),
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