On K3 correspondences

Khalid, Madeeha

doi:10.1515/crll.2005.2005.589.57

Cited by 5 publications

(3 citation statements)

References 4 publications

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“…In general M is not a fine moduli space, that is a universal sheaf does not exist. However a quasi-universal sheaf E does exist such that E | X×m F m 2 where F m is a representative of the isomorphism class of bundles corresponding to m [15].…”

Section: Which Sends a Maximal Isotropic Subspace To The Ideal It Genmentioning

confidence: 99%

An explicit derived equivalence of Azumaya algebras on K3 surfaces via Koszul duality

Ingalls

Khalid

2015

Journal of Algebra

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We consider moduli spaces of Azumaya algebras on K3 surfaces and construct an example. In some cases we show a derived equivalence which corresponds to a derived equivalence between twisted sheaves. We prove if A and A are Morita equivalent Azumaya algebras of degree r then 2r divides c 2 (A) − c 2 (A ). In particular this implies that if A is an Azumaya algebra on a K3 surface and c 2 (A) is within 2r of its minimal bound then the moduli stack of Azumaya algebras with the same underlying gerbe, if non-empty, is a proper algebraic space.

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Section: Which Sends a Maximal Isotropic Subspace To The Ideal It Genmentioning

confidence: 99%

An explicit derived equivalence of Azumaya algebras on K3 surfaces via Koszul duality

Ingalls

Khalid

2015

Journal of Algebra

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“…In fact, the conjecture could be verified in a number of other situations (see e.g. [14,35]), but turned out to be wrong in general (see [30,Ex. 4.11]).…”

Section: Any Fourier-mukai Transform φmentioning

confidence: 99%

The global Torelli theorem: classical, derived, twisted

Huybrechts¹

2009

Algebraic Geometry

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These notes survey work on various aspects and generalizations of the Global Torelli Theorem for K3 surfaces done over the last ten years. The classical Global Torelli Theorem was proved a long time ago (see [9,39,52,57]), but the interest in similar questions has been revived by the new approach to K3 surfaces suggested by mirror symmetry.Kontsevich proposed to view mirror symmetry as an equivalence between the bounded derived category of coherent sheaves on a Calabi-Yau manifold and the derived Fukaya category of its mirror dual. For an algebraic geometer the bounded derived category of coherent sheaves on a variety is a familiar object, but to view it as an interesting invariant of the variety rather than a technical tool to deal with cohomology is rather surprising. Due to results of Mukai, Orlov, and Polishchuk, the bounded derived category of coherent sheaves on an abelian variety is completely understood, i.e. we know when two abelian varieties yield equivalent derived categories and what the group of autoequivalences looks like.Independently of their importance in mirror symmetry, K3 surfaces form the next most simple class of Calabi-Yau manifolds and one would like to study them from a derived category point of view, too. The program has been started already by Mukai in [43] and the break-through came with Orlov's article [48]. But this was still not the end. For many reasons (mirror symmetry considerations, existence of non-fine moduli spaces, geometric interpretation of conformal field theories, etc.) one would like to incorporate B-fields or, closely related, Brauer classes in the picture. These notes will mostly concentrate on aspects that are related to generalizations of the Global Torelli Theorem in this direction. The following list of topics gives an idea of what shall be discussed:• Hitchin's generalized Calabi-Yau structures.• The period of a twisted K3 surface.• Brauer group and B-fields.• Derived categories of twisted sheaves.This survey contains essentially no proofs. I have tried to emphasize ideas and refer to the literature for details. Some of the results can very naturally be viewed in terms of moduli spaces of generalized K3 surfaces and the action of the mirror symmetry group, but I have decided to neglect these aspects almost completely. Although mirror symmetry has shaped our way of thinking about derived categories, the symplectic side of the theory will not be touched upon.

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“…there does not exist a universal sheaf E on X × M such that E | X×m ≃ E m where E m is the isomorphism class of sheaves corresponding to m in M. However a quasi-universal sheaf does exist. An explicit construction of a quasi-universal sheaf is in [13].…”

Section: Introductionmentioning

confidence: 99%

Rank 2 Sheaves on K3 Surfaces: A Special Construction

Ingalls¹,

Khalid²

2012

The Quarterly Journal of Mathematics

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Abstract. Let X be a K3 surface of degree 8 in P 5 with hyperplane section H. We associate to it another K3 surface M which is a double cover of P 2 ramified on a sextic curve C. In the generic case when X is smooth and a complete intersection of three quadrics, there is a natural correspondence between M and the moduli space M of rank two vector bundles on X with Chern classes c 1 = H and c 2 = 4. We build on previous work of Mukai and others, giving conditions and examples where M is fine, compact, non-empty; and birational or isomorphic to M . We also present an explicit calculation of the Fourier-Mukai transform when X contains a line and has Picard number two.

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On K3 correspondences

Cited by 5 publications

References 4 publications

An explicit derived equivalence of Azumaya algebras on K3 surfaces via Koszul duality

An explicit derived equivalence of Azumaya algebras on K3 surfaces via Koszul duality

The global Torelli theorem: classical, derived, twisted

Rank 2 Sheaves on K3 Surfaces: A Special Construction

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