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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 150 (5 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
Braid group actions on derived categories of coherent sheaves
 Duke Math. J
"... This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety X. The motivation for this is M. Kontsevich’s homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is ..."
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Cited by 136 (7 self)
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This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety X. The motivation for this is M. Kontsevich’s homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dim X ≥ 2, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail. 1.
Hypergeometric functions and mirror symmetry in toric varieties
, 1999
"... We study aspects related to Kontsevich’s homological mirror symmetry conjecture [42] in the case of Calabi–Yau complete intersections in toric varieties. In a 1996 lecture, Kontsevich [43] indicated how his proposal implies that the groups of automorphisms of the two types of categories involved in ..."
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Cited by 56 (4 self)
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We study aspects related to Kontsevich’s homological mirror symmetry conjecture [42] in the case of Calabi–Yau complete intersections in toric varieties. In a 1996 lecture, Kontsevich [43] indicated how his proposal implies that the groups of automorphisms of the two types of categories involved in the homological mirror symmetry conjecture should also be identified. Our results provide an explicit geometric construction of the correspondence between the automorphisms of the two types of categories. We compare the monodromy calculations for the Picard–Fuchs system associated with the periods of a Calabi–Yau manifold M with the algebrogeometric computations of the cohomology action of Fourier– Mukai functors on the bounded derived category of coherent sheaves on the mirror Calabi–Yau manifold W. We obtain the complete dictionary between the two sides for the one complex parameter case of Calabi–Yau complete intersections in weighted projective spaces, as well as for some two parameter cases. We also find the complex of sheaves on W × W that corresponds to a loop in the moduli space of complex structures on M induced by a phase transition of W.
derived categories, and Grothendieck groups,” Nucl. Phys. B561
, 1999
"... In this paper we describe how Grothendieck groups of coherent sheaves and locally free sheaves can be used to describe type II Dbranes, in the case that all Dbranes are wrapped on complex varieties and all connections are holomorphic. Our proposal is in the same spirit as recent discussions of Kt ..."
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Cited by 44 (12 self)
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In this paper we describe how Grothendieck groups of coherent sheaves and locally free sheaves can be used to describe type II Dbranes, in the case that all Dbranes are wrapped on complex varieties and all connections are holomorphic. Our proposal is in the same spirit as recent discussions of Ktheory and Dbranes; within the restricted class mentioned, Grothendieck groups encode a choice of connection on each Dbrane worldvolume, in addition to information about the C ∞ bundles. We also point out that derived categories can also be used to give insight into Dbrane constructions, and analyze how a Z2 subset of the Tduality group acting on Dbranes on tori can be understood in terms of a FourierMukai transformation.
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 44 (2 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
Derived Category Automorphisms from Mirror Symmetry
 Duke Math. J
"... Inspired by the homological mirror symmetry conjecture of Kontsevich [24], we construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth Calabi–Yau variety. 1 ..."
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Cited by 38 (0 self)
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Inspired by the homological mirror symmetry conjecture of Kontsevich [24], we construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth Calabi–Yau variety. 1
Dbranes on CalabiYau manifolds
, 2004
"... In this review we study BPS Dbranes on Calabi–Yau threefolds. Such Dbranes naturally divide into two sets called Abranes and Bbranes which are most easily understood from topological field theory. The main aim of this paper is to provide a selfcontained guide to the derived category approach to ..."
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Cited by 36 (7 self)
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In this review we study BPS Dbranes on Calabi–Yau threefolds. Such Dbranes naturally divide into two sets called Abranes and Bbranes which are most easily understood from topological field theory. The main aim of this paper is to provide a selfcontained guide to the derived category approach to Bbranes and the idea of Πstability. We argue that this mathematical machinery is hard to avoid for a proper understanding of Bbranes. Abranes and Bbranes are related in a very complicated and interesting way which ties in with the “homological mirror symmetry ” conjecture of Kontsevich. We motivate and exploit this form of mirror symmetry. The examples of the quintic 3fold, flops and orbifolds are discussed at some length. In the latter
FourierMukai transforms for K3 and elliptic fibrations
"... Abstract. Given a nonsingular variety with a K3 fibration π: X → S we construct dual fibrations ˆπ: Y → S by replacing each fibre Xs of π by a twodimensional moduli space of stable sheaves on Xs. In certain cases we prove that the resulting scheme Y is a nonsingular variety and construct an equiv ..."
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Cited by 36 (4 self)
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Abstract. Given a nonsingular variety with a K3 fibration π: X → S we construct dual fibrations ˆπ: Y → S by replacing each fibre Xs of π by a twodimensional moduli space of stable sheaves on Xs. In certain cases we prove that the resulting scheme Y is a nonsingular variety and construct an equivalence of derived categories of coherent sheaves Φ: D(Y) → D(X). Our methods also apply to elliptic and abelian surface fibrations. As an application we show how the equivalences Φ identify certain moduli spaces of stable bundles on elliptic threefolds with Hilbert schemes of curves. 1.