Global matrix factorizations

Lin, Kevin; Pomerleano, Daniel

doi:10.48550/arxiv.1101.5847

Cited by 13 publications

(34 citation statements)

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“…There are several versions of this notion; the appropriate one for quasi-coherent sheaves is called the coderived category and for coherent sheaves it is the absolute derived category. The absolute derived category of locally free matrix factorizations was studied in [37]; for coherent matrix factorizations over a smooth variety, it was considered in [26]. These two absolute derived categories are equivalent for regular schemes, but can be different otherwise (as we show with an explicit counterexample).…”

Section: Introductionmentioning

confidence: 91%

“…Following [26], we will consider CDG-modules over B = (X, L, w) whose underlying graded B-modules correspond to coherent or quasi-coherent O X -modules, rather than just locally free sheaves (as in the conventional matrix factorizations). A quasicoherent CDG-module over (X, L, w) is the same thing as a pair of quasi-coherent…”

Section: Matrix Factorizationsmentioning

confidence: 99%

“…Another name for this notion is "D-branes in the Landau-Ginzburg B model" (as suggested by Kontsevich) [24]; in this context, the element w is called the potential. One generalizes the above definition, replacing free modules with projective modules [24,34], with locally free sheaves [37], and finally with coherent sheaves [26]. The importance of the latter generalization is emphasized in the present paper.…”

Section: Introductionmentioning

confidence: 97%

“…Acknowledgment. The second author is grateful to Daniel Pomerleano and Kevin Lin for sending him a preliminary version of their manuscript [26], which inspired the present work. We also wish to thank Roman Bezrukavnikov, Alexander Kuznetsov, and Amnon Neeman for helpful conversations, Alexander Polishchuk and Henning Krause for stimulating questions, and Dmitri Orlov for his insightful remarks on an early version of this manuscript.…”

Section: Introductionmentioning

confidence: 98%

“…A version of (the former two of) these results is used in Section 2 in order to extend Orlov's cokernel functor from the absolute derived category of locally free matrix factorizations to the absolute derived category of coherent ones. This extension of the cokernel functor admits a simple construction of a functor in the opposite direction, suggested in [26]. We use these constructions to obtain a new proof of Orlov's theorem, and our own generalization of it to the singular case.…”

Section: Introductionmentioning

confidence: 99%

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Coherent analogues of matrix factorizations and relative singularity categories

Efimov

Positselski

2015

Algebra Number Theory

116

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Abstract. We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations is then identified with the triangulated category of relative singularities, while the similar exotic derived category of locally free matrix factorizations is its full subcategory. The latter category is identified with the kernel of the direct image functor corresponding to the closed embedding of the zero locus and acting between the conventional (absolute) triangulated categories of singularities. Similar results are obtained for matrix factorizations of infinite rank; and two different "large" versions of the triangulated category of relative singularities, corresponding to the approaches of Orlov and Krause, are identified in the case of a Cartier divisor. A version of the Thomason-Trobaugh-Neeman localization theory is proven for coherent matrix factorizations and disproven for locally free matrix factorizations of finite rank. Contravariant (coherent) and covariant (quasi-coherent) versions of the Serre-Grothendieck duality theorems for matrix factorizations are established, and pull-backs and push-forwards of matrix factorizations are discussed at length. A number of general results about derived categories of the second kind for CDG-modules over quasi-coherent CDG-algebras are proven on the way. Hochschild (co)homology of matrix factorization categories are discussed in an appendix.

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Section: Introductionmentioning

confidence: 91%

Section: Matrix Factorizationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Coherent analogues of matrix factorizations and relative singularity categories

Efimov

Positselski

2015

Algebra Number Theory

116

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Matrix factorizations for nonaffine LG–models

Orlov¹

2011

Math. Ann.

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A category of kernels for equivariant factorizations and its implications for Hodge theory

2013

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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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Global matrix factorizations

Cited by 13 publications

References 0 publications

Coherent analogues of matrix factorizations and relative singularity categories

Coherent analogues of matrix factorizations and relative singularity categories

Matrix factorizations for nonaffine LG–models

A category of kernels for equivariant factorizations and its implications for Hodge theory

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