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22
Cyclic homology, cdhcohomology and negative Ktheory
, 2005
"... We prove a blowup formula for cyclic homology which we use to show that infinitesimal Ktheory satisfies cdhdescent. Combining that result with some computations of the cdhcohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic Ktheor ..."
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Cited by 37 (12 self)
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We prove a blowup formula for cyclic homology which we use to show that infinitesimal Ktheory satisfies cdhdescent. Combining that result with some computations of the cdhcohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic Ktheory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.
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 29 (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.
ON SERRE DUALITY FOR COMPACT HOMOLOGICALLY SMOOTH DG ALGEBRAS
, 2007
"... Let X be a smooth projective variety over a perfect field k. It is a classical fact that the ..."
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Cited by 16 (2 self)
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Let X be a smooth projective variety over a perfect field k. It is a classical fact that the
Orbifold Cohomology as Periodic Cyclic Homology
, 2002
"... It known from the work of FeiginTsygan, Weibel and Keller that the cohomology groups of a smooth complex variety X can be recovered from (roughly speaking) its derived category of coherent sheaves. In this paper we show that for a finite group G acting on X the same procedure applied to Gequivaria ..."
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Cited by 15 (1 self)
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It known from the work of FeiginTsygan, Weibel and Keller that the cohomology groups of a smooth complex variety X can be recovered from (roughly speaking) its derived category of coherent sheaves. In this paper we show that for a finite group G acting on X the same procedure applied to Gequivariant sheaves gives the orbifold cohomology of X/G. As an application, in some cases we are able to obtain simple proofs of an additive isomorphism between the orbifold cohomology of X/G and the usual cohomology of its crepant resolution (the equality of Euler and Hodge numbers was obtained earlier by various authors). We also state some conjectures on the product structures, as well as the singular case; and a connection with a recent work by Kawamata.
The Mukai pairing and integral transforms in Hochschild homology
"... Let X be a smooth proper scheme over a field of characteristic 0. Following Shklyarov [10] , we construct a (nondegenerate) pairing on the Hochschild homology of perf (X), and hence, on the Hochschild homology of X. On the other hand the Hochschild homology of X also has the Mukai pairing (see [1]) ..."
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Cited by 13 (1 self)
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Let X be a smooth proper scheme over a field of characteristic 0. Following Shklyarov [10] , we construct a (nondegenerate) pairing on the Hochschild homology of perf (X), and hence, on the Hochschild homology of X. On the other hand the Hochschild homology of X also has the Mukai pairing (see [1]). If X is CalabiYau, this pairing arises from the action of the class of a genus 0 Riemannsurface with two incoming closed boundaries and no outgoing boundary in H0(M0(2, 0)) on the algebra of closed states of a version of the BModel on X. We show that these pairings ”almost ” coincide. This is done via a different view of the construction of integral transforms in Hochschild homology that originally appeared in Caldararu’s work [1]. This is used to prove that the more ”natural ” construction of integral transforms in Hochschild homology by Shklyarov [10] coincides with that of Caldararu [1]. These results give rise to a Hirzebruch RiemannRoch
THE GROMOVWITTEN POTENTIAL ASSOCIATED TO A TCFT
, 2005
"... Abstract. This is the sequel to my preprint“TCFTs and CalabiYau categories”. Here we extend the results of that paper to construct, for certain CalabiYau A∞ categories, something playing the role of the GromovWitten potential. This is a state in the Fock space associated to periodic cyclic homolo ..."
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Cited by 8 (2 self)
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Abstract. This is the sequel to my preprint“TCFTs and CalabiYau categories”. Here we extend the results of that paper to construct, for certain CalabiYau A∞ categories, something playing the role of the GromovWitten potential. This is a state in the Fock space associated to periodic cyclic homology, which is a symplectic vector space. Applying this to a suitable A ∞ version of the derived category of sheaves on a CalabiYau yields the B model potential, at all genera. The construction doesn’t go via the DeligneMumford spaces, but instead uses the BatalinVilkovisky algebra constructed from the uncompactified moduli spaces of curves by Sen and Zwiebach. The fundamental class of DeligneMumford space is replaced here by a certain solution of the quantum master equation, essentially the “string vertices ” of Zwiebach. On the field theory side, the BV operator has an interpretation as the quantised differential on the Fock space for periodic cyclic chains. Passing to homology, something satisfying the master equation yields an element of the Fock space. 1. Notation We work throughout over a ground field K containing Q. Often we will use topological K vector spaces. All tensor products will be completed. All the topological vector spaces we use are inverse limits, so the completed tensor product is also an inverse limit. All the results remain true without any change if we work over a differential graded ground ring R, and use flat R modules. (An R module is flat if the functor of tensor product with it is exact). We could also have only a Z/2 grading on R. 2. Acknowledgements I would like to thank Tom Coates, Ezra Getzler, Alexander Givental and Paul Seidel for very helpful conversations, and Dennis Sullivan for explaining to me his ideas on the BatalinVilkovisky formalism and moduli spaces of curves. 3. Topological conformal field theories Let S be the topological category whose objects are the nonnegative integers, and whose morphism space S(n,m) is the moduli space of Riemann surfaces with n parameterised incoming and m parameterised outgoing boundaries, such that each connected component has at least one incoming boundary. These surfaces are not necessarily connected. Let Sχ(n,m) ⊂ S(n,m) be the space of surfaces of Euler characteristic χ.
Perverse tstructure on Artin stacks
 Math. Z
"... Abstract. In this paper we develop the theory of perverse sheaves on Artin stacks continuing the study in [12] and [13]. 1. ..."
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Cited by 8 (0 self)
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Abstract. In this paper we develop the theory of perverse sheaves on Artin stacks continuing the study in [12] and [13]. 1.
CLOSED STRING TCFT FOR HERMITIAN CALABIYAU ELLIPTIC SPACES
, 807
"... Abstract. We describe an explicit action of the prop of the chains on the moduli space of Riemann surfaces on the Hochschild complex of a CalabiYau elliptic space. One example of such an elliptic space extends the known string topology operations, for all compact simplyconnected manifolds, to a col ..."
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Cited by 5 (2 self)
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Abstract. We describe an explicit action of the prop of the chains on the moduli space of Riemann surfaces on the Hochschild complex of a CalabiYau elliptic space. One example of such an elliptic space extends the known string topology operations, for all compact simplyconnected manifolds, to a collection indexed by the de Rham currents on the moduli space. Another example pertains to the Bmodel at all genera. Contents