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157
The McKay correspondence as an equivalence of derived categories
 J. Amer. Math. Soc
"... The classical McKay correspondence relates representations of a nite subgroup G SL(2;C) to the cohomology of the wellknown minimal resolution of the Kleinian singularity C2=G. GonzalezSprinberg and Verdier [10] interpreted the McKay correspondence as an isomorphism on K theory, observing that the ..."
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Cited by 239 (6 self)
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The classical McKay correspondence relates representations of a nite subgroup G SL(2;C) to the cohomology of the wellknown minimal resolution of the Kleinian singularity C2=G. GonzalezSprinberg and Verdier [10] interpreted the McKay correspondence as an isomorphism on K theory, observing that the repre
Generators and representability of functors in commutative and noncommutative geometry
 MOSC MATH. J
, 2002
"... We give a sufficient condition for an Extfinite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived ca ..."
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Cited by 205 (4 self)
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We give a sufficient condition for an Extfinite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, hence saturated. In contrast the similar category for a smooth compact analytic surface with no curves is not saturated.
SEMIORTHOGONAL DECOMPOSITIONS FOR ALGEBRAIC VARIETIES
, 1995
"... A criterion for a functor between derived categories of coherent sheaves to be full and faithful is given. A semiorthogonal decomposition for the derived category of coherent sheaves on the intersection of two even dimensional quadrics is obtained. The behaviour of derived categories with respect to ..."
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Cited by 181 (11 self)
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A criterion for a functor between derived categories of coherent sheaves to be full and faithful is given. A semiorthogonal decomposition for the derived category of coherent sheaves on the intersection of two even dimensional quadrics is obtained. The behaviour of derived categories with respect to birational transformations is investigated. A theorem about reconstruction of a variety from the
Derived categories of coherent sheaves and triangulated categories of singularities
, 2005
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Equivalences of triangulated categories and FourierMukai transforms
 Bull. London Math. Soc
, 1999
"... Abstract. We give a condition for an exact functor between triangulated categories to be an equivalence. Applications to FourierMukai transforms are discussed. In particular we obtain a large number of such transforms for K3 surfaces. 1. ..."
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Cited by 123 (7 self)
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Abstract. We give a condition for an exact functor between triangulated categories to be an equivalence. Applications to FourierMukai transforms are discussed. In particular we obtain a large number of such transforms for K3 surfaces. 1.
Mirror symmetry for weighted projective planes and their noncommutative deformations
, 2004
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Flops and derived categories
 Invent. Math
"... This paper contains some applications of FourierMukai techniques to problems in birational geometry. The main new idea is that flops occur naturally as moduli ..."
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Cited by 89 (3 self)
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This paper contains some applications of FourierMukai techniques to problems in birational geometry. The main new idea is that flops occur naturally as moduli
FourierMukai transforms for elliptic surfaces
, 1998
"... Abstract. We compute a large number of moduli spaces of stable bundles on a general algebraic elliptic surface using a new class of relative FourierMukai transforms. 1. ..."
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Cited by 67 (4 self)
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Abstract. We compute a large number of moduli spaces of stable bundles on a general algebraic elliptic surface using a new class of relative FourierMukai transforms. 1.
tstructures on some local CalabiYau varieties
 J. Algebra
"... Abstract. Let Z be a Fano varity satisfying the condition that the rank of the Grothendieck group of Z is one more than the dimension of Z. Let ωZ denote the total space of the canonical line bundle of Z, considered as a noncompact CalabiYau variety. We use the theory of exceptional collections to ..."
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Cited by 59 (3 self)
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Abstract. Let Z be a Fano varity satisfying the condition that the rank of the Grothendieck group of Z is one more than the dimension of Z. Let ωZ denote the total space of the canonical line bundle of Z, considered as a noncompact CalabiYau variety. We use the theory of exceptional collections to describe tstructures on the derived category of coherent sheaves on ωZ. The combinatorics of these tstructures is determined by a natural action of an affine braid group, closely related to the wellknown action of the Artin braid group on the set of exceptional collections on Z. 1.
Derived Categories of Quadric Fibrations and Intersections of Quadrics
, 2005
"... We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algeb ..."
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Cited by 52 (12 self)
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We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algebras on the base corresponding to this quadric fibration generalizing the Kapranov’s description of the derived category of a single quadric. As an application we verify that the noncommutative algebraic variety (P(S 2 W ∗), B0), where B0 is the universal sheaf of even parts of Clifford algebras, is Homologically Projectively Dual to the projective space P(W) in the double Veronese embedding P(W) → P(S 2 W). Using the properties of the Homological Projective Duality we obtain a description of the derived category of coherent sheaves on a complete intersection of any number of quadrics.