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235
Equivariant Cohomology, Koszul Duality, and the Localization Theorem
 Invent. Math
, 1998
"... This paper concerns three aspects of the action of a compact group K on a space ..."
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Cited by 147 (4 self)
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This paper concerns three aspects of the action of a compact group K on a space
Noncommutative FiniteDimensional Manifolds  I. SPHERICAL MANIFOLDS AND RELATED EXAMPLES
, 2001
"... We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 d ..."
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Cited by 89 (12 self)
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We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 defined by basic Ktheoretic equations. We find a 3parameter family of deformations of the standard 3sphere S 3 and a corresponding 3parameter deformation of the 4dimensional Euclidean space R 4. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R 4 u are isomorphic to the algebras introduced by Sklyanin in connection with the YangBaxter equation. Special values of the deformation parameters do not give rise to Sklyanin algebras and we extract a subclass, the θdeformations, which we generalize in any dimension and various contexts, and study in some details. Here, and
Leung : The Enumerative Geometry of K3 Surfaces and Modular Forms, eprint alggeom 9711031
"... Abstract. Let X be a K3 surface and C be a holomorphic curve in X representing a primitive homology class. We count the number of curves of geometric genus g with n nodes passing through g generic points in X in the linear system C  for any g and n satisfying C · C = 2g + 2n − 2. When g = 0, this ..."
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Cited by 45 (8 self)
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Abstract. Let X be a K3 surface and C be a holomorphic curve in X representing a primitive homology class. We count the number of curves of geometric genus g with n nodes passing through g generic points in X in the linear system C  for any g and n satisfying C · C = 2g + 2n − 2. When g = 0, this coincides with the enumerative problem studied by Yau and Zaslow who obtained a conjectural generating function for the numbers. Recently, Göttsche has generalized their conjecture to arbitrary g in terms of quasimodular forms. We prove these formulas using GromovWitten invariants for families, a degeneration argument, and an obstruction bundle computation. Our methods also apply to P 2 blown up at 9 points where we show that the ordinary GromovWitten invariants of genus g constrained to g points are also given in terms of quasimodular forms. Contents
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.
THE SPECTRAL SEQUENCE RELATING ALGEBRAIC KTHEORY TO MOTIVIC COHOMOLOGY
"... The purpose of this paper is to establish in Theorem 13.13 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic Ktheory of X: E p,q 2 = Hp−q (X, Z(−q)) = CH −q (X, −p − q) ⇒ K−p−q(X). (13.13.1) Such a spectral sequence was conjectured by A. Beilins ..."
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Cited by 44 (5 self)
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The purpose of this paper is to establish in Theorem 13.13 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic Ktheory of X: E p,q 2 = Hp−q (X, Z(−q)) = CH −q (X, −p − q) ⇒ K−p−q(X). (13.13.1) Such a spectral sequence was conjectured by A. Beilinson [Be] as a natural analogue of the AtiyahHirzebruch spectral sequence from the singular cohomology to the topological Ktheory of a topological space. The expectation of such a spectral sequence has provided much of the impetus for the development of motivic cohomology (e.g., [B1], [V2]) and should facilitate many computations in algebraic Ktheory. In the special case in which X equals SpecF, this spectral sequence was established by S. Bloch and S. Lichtenbaum [BL]. Our construction depends crucially upon the main result of [BL], the existence of an exact couple relating the motivic cohomology of the field F to the multirelative Ktheory of coherent sheaves on standard simplices over F (recalled as Theorem 5.5 below). A major step in generalizing the work of Bloch and Lichtenbaum is our reinterpretation of their spectral sequence in terms of the “topological filtration ” on the Ktheory of the standard cosimplicial scheme ∆ • over F. We find that the spectral sequence arises from a tower of Ωprespectra K( ∆ • ) = K 0 ( ∆ • ) ← − K 1 ( ∆ • ) ← − K 2 ( ∆ • ) ← − · · · Thus, even in the special case in which X equals SpecF, we obtain a much clearer understanding of the BlochLichtenbaum spectral sequence which is essential for purposes of generalization. Following this reinterpretation, we proceed using techniques introduced by V. Voevodsky in his study of motivic cohomology. In order to do this, we provide an equivalent formulation of Ktheory spectra associated to coherent sheaves on X with conditions on their supports K q ( ∆ • × X) which is functorial in X. We then Partially supported by the N.S.F. and the N.S.A.
KazhdanLusztig polynomials and character formulae for the Lie superalgebra gl(mn
 J. AMS
"... The problem of computing the characters of the finite dimensional irreducible representations of the Lie superalgebra gl(mn) over C was solved a few years ago by V. Serganova [19]. In this article, we present an entirely different approach. We also formulate a precise ..."
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Cited by 42 (5 self)
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The problem of computing the characters of the finite dimensional irreducible representations of the Lie superalgebra gl(mn) over C was solved a few years ago by V. Serganova [19]. In this article, we present an entirely different approach. We also formulate a precise
Combinatorics Of Branchings In Higher Dimensional Automata
 Theory Appl. Categ
, 2001
"... We explore the combinatorial properties of the branching areas of execution paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner (or branching) homology of a globular #category and the combinatorics of a new homology theory ca ..."
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Cited by 35 (9 self)
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We explore the combinatorial properties of the branching areas of execution paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner (or branching) homology of a globular #category and the combinatorics of a new homology theory called the reduced branching homology. The latter is the homology of the quotient of the branching complex by the subcomplex generated by its thin elements. Conjecturally it coincides with the non reduced theory for higher dimensional automata, that is #categories freely generated by precubical sets. As application, we calculate the branching homology of some #categories and we give some invariance results for the reduced branching homology. We only treat the branching side. The merging side, that is the case of merging areas of execution paths is similar and can be easily deduced from the branching side.
Complete modules and torsion modules
 Amer. J. Math
"... Abstract. Suppose that R is a ring and that A is a chain complex over R. Inside the derived category of differential graded Rmodules there are naturally defined subcategories of Atorsion objects and of Acomplete objects. Under a finiteness condition on A, we develop a Morita theory for these subc ..."
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Cited by 34 (5 self)
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Abstract. Suppose that R is a ring and that A is a chain complex over R. Inside the derived category of differential graded Rmodules there are naturally defined subcategories of Atorsion objects and of Acomplete objects. Under a finiteness condition on A, we develop a Morita theory for these subcategories, find conceptual interpretations for some associated algebraic functors, and, in appropriate commutative situations, identify the associated functors as local homology or local cohomology. Some of the results are suprising even in the case R = Z and A = Z/p. 1.
Homotopical Algebraic Geometry I: Topos theory
, 2002
"... This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ..."
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Cited by 32 (20 self)
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This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ∞categories, and we develop the notions of Stopologies, Ssites and stacks over them. We prove in particular, that for an Scategory T endowed with an Stopology, there exists a model