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HOMOLOGICAL PROJECTIVE DUALITY FOR GRASSMANNIANS OF LINES
, 2006
"... Abstract. We show that homologically projectively dual varieties for Grassmannians Gr(2, 6) and Gr(2,7) are given by certain noncommutative resolutions of singularities of the corresponding Pfaffian varieties. As an application we describe the derived categories of linear sections of these Grassmann ..."
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Cited by 38 (8 self)
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Abstract. We show that homologically projectively dual varieties for Grassmannians Gr(2, 6) and Gr(2,7) are given by certain noncommutative resolutions of singularities of the corresponding Pfaffian varieties. As an application we describe the derived categories of linear sections of these Grassmannians and Pfaffians. In particular, we show that
Hochschild homology and semiorthogonal decompositions
"... Abstract. We investigate Hochschild cohomology and homology of admissible subcategories of derived categories of coherent sheaves on smooth projective varieties. We show that the Hochschild cohomology of an admissible subcategory is isomorphic to the derived endomorphisms of the kernel giving the co ..."
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Cited by 28 (2 self)
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Abstract. We investigate Hochschild cohomology and homology of admissible subcategories of derived categories of coherent sheaves on smooth projective varieties. We show that the Hochschild cohomology of an admissible subcategory is isomorphic to the derived endomorphisms of the kernel giving the corresponding projection functor, and the Hochschild homology is isomorphic to derived morphisms from this kernel to its convolution with the kernel of the Serre functor. We investigate some basic properties of Hochschild homology and cohomology of admissible subcategories. In particular, we check that the Hochschild homology is additive with respect to semiorthogonal decompositions and construct some long exact sequences relating the Hochschild cohomology of a category and its semiorthogonal components. We also compute Hochschild homology and cohomology of some interesting admissible subcategories, in particular of the nontrivial components of derived categories of some Fano threefolds and of the nontrivial components of the derived categories of conic bundles. 1.
NonBirational Twisted Derived Equivalences in Abelian GLSMs
, 2007
"... In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between nonbirational spaces, and realizing geometries in novel fashions. Examples of gauged linear sigma models with nonbirational Kähler phases are a relatively new phenomenon. Mos ..."
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Cited by 23 (9 self)
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In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between nonbirational spaces, and realizing geometries in novel fashions. Examples of gauged linear sigma models with nonbirational Kähler phases are a relatively new phenomenon. Most of our examples involve gauged linear sigma models for complete intersections of quadric hypersurfaces, though we also discuss some more general cases and their interpretation. We also propose a more general understanding of the relationship between Kähler phases of gauged linear sigma models, namely that they are related by (and realize) Kuznetsov’s ‘homological projective duality. ’ Along the way, we shall see how ‘noncommutative spaces ’ (in Kontsevich’s sense) are realized physically in gauged linear sigma models, providing examples of new types of conformal field theories. Throughout, the physical realization of stacks plays a key role in interpreting physical structures appearing in GLSMs, and we find that stacks are implicitly much more common in GLSMs than previously realized.
HOMOLOGICAL PROJECTIVE DUALITY
, 2005
"... Abstract. We introduce a notion of Homological Projective Duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties X and Y in dual projective spaces are Homologically Projectively Dual, then we prove that th ..."
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Cited by 22 (6 self)
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Abstract. We introduce a notion of Homological Projective Duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties X and Y in dual projective spaces are Homologically Projectively Dual, then we prove that the orthogonal linear sections of X and Y admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singularities of these sections are equivalent. We also investigate Homological Projective Duality for projectivizations of vector bundles. 1.
Derived categories of coherent sheaves
 Proceedings of the 2006 ICM
, 2006
"... Abstract. We discuss derived categories of coherent sheaves on algebraic varieties. We focus on the case of nonsingular CalabiYau varieties and consider two unsolved problems: proving that birational varieties have equivalent derived categories, and computing the group of derived autoequivalences. ..."
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Cited by 16 (0 self)
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Abstract. We discuss derived categories of coherent sheaves on algebraic varieties. We focus on the case of nonsingular CalabiYau varieties and consider two unsolved problems: proving that birational varieties have equivalent derived categories, and computing the group of derived autoequivalences. We also introduce the space of stability conditions on a triangulated category and explain its relevance to these two problems. 1.
Duality in twodimensional (2,2) supersymmetric nonAbelian gauge theories,” arXiv:1104.2853 [hepth
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A CATEGORICAL INVARIANT FOR CUBIC THREEFOLDS
, 903
"... Abstract. We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the nontrivial part of a semiorthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class. ..."
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Cited by 11 (4 self)
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Abstract. We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the nontrivial part of a semiorthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class.
Some remarks on the derived categories of coherent sheaves on homogeneous spaces
 ALGEBRA SECTION, STEKLOV MATHEMATICAL INSTITUTE, 8 GUBKIN STR., MOSCOW 119991 RUSSIA EMAIL ADDRESS: AKUZNET@MI.RAS.RU
, 2006
"... In this paper we prove first a general theorem on semiorthogonal decompositions in derived categories of coherent sheaves for flat families over a smooth base. We then show that the derived categories of coherent sheaves on flag varieties of classical type are generated by complete exceptional colle ..."
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Cited by 10 (0 self)
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In this paper we prove first a general theorem on semiorthogonal decompositions in derived categories of coherent sheaves for flat families over a smooth base. We then show that the derived categories of coherent sheaves on flag varieties of classical type are generated by complete exceptional collections. Finally, we find complete exceptional collections in the derived categories of some homogeneous spaces of the symplectic groups of small rank.
Derived categories of Fano threefolds
 arXiv:0809.0225. �� 0. CATEGORICAL INVARIANT FOR CUBIC THREEFOLDS 23
"... Abstract. We consider the structure of the derived categories of coherent sheaves on Fano threefolds with Picard number 1 and describe a strange relation between derived categories of different threefolds. In the Appendix we discuss how the ring of algebraic cycles of a smooth projective variety is ..."
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Cited by 10 (2 self)
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Abstract. We consider the structure of the derived categories of coherent sheaves on Fano threefolds with Picard number 1 and describe a strange relation between derived categories of different threefolds. In the Appendix we discuss how the ring of algebraic cycles of a smooth projective variety is related to the Grothendieck group of its derived category. 1.