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45
Roitman’s Theorem for Singular Complex Projective Surfaces
"... Let X be a complex projective surface with arbitrary singularities. We construct a generalized Abel–Jacobi map A0(X) → J 2 (X) and show that it is an isomorphism on torsion subgroups. Here A0(X) is the appropriate Chow group of smooth 0cycles of degree 0 on X, and J 2 (X) is the intermediate Jacob ..."
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Let X be a complex projective surface with arbitrary singularities. We construct a generalized Abel–Jacobi map A0(X) → J 2 (X) and show that it is an isomorphism on torsion subgroups. Here A0(X) is the appropriate Chow group of smooth 0cycles of degree 0 on X, and J 2 (X) is the intermediate Jacobian associated with the mixed Hodge structure on H 3 (X). Our result generalizes a theorem of Roitman for smooth surfaces: if X is smooth then the torsion in the usual Chow group A0(X) is isomorphic to the torsion in the usual Albanese variety J 2 (X) ∼ = Alb(X) by the classical AbelJacobi map.
Comparing Ktheories for complex varieties
, 2000
"... Abstract. The semitopological Ktheory of a complex variety was defined in [FW2] with the expectation that it would prove to be a theory lying “part way ” between the algebraic Ktheory of the variety and the topological Ktheory of the associated analytic space, and thus would share properties wi ..."
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Abstract. The semitopological Ktheory of a complex variety was defined in [FW2] with the expectation that it would prove to be a theory lying “part way ” between the algebraic Ktheory of the variety and the topological Ktheory of the associated analytic space, and thus would share properties with each of these other theories. In this paper, we realize these expectations by proving among other results that (1) the algebraic Ktheory with finite coefficients and the semitopological Ktheory with finite coefficients coincide on all projective complex varieties, (2) semitopological Ktheory and topological Ktheory agree on certain types of generalized flag varieties, and (3) (by building on a result of Cohen and LimaFilho) the semitopological Ktheory of any smooth projective variety becomes isomorphic to the topological Ktheory of the underlying analytic space once the Bott element is inverted. To illustrate the utility of our results, we observe that a new proof of the QuillenLichtenbaum conjecture for smooth, complete curves is obtained as a corollary. In the recent paper [FW2], the authors introduced “semitopological Ktheory” Ksemi ∗ (X) for a complex quasiprojective algebraic variety X, showed that the natural
BLOCHOGUS PROPERTIES FOR TOPOLOGICAL CYCLE THEORY
"... In this paper, we reformulate “morphic cohomology ” as introduced by the author and H.B. Lawson [FL1] in such a way that it and “Lawson homology ” satisfy the list of basic properties codified by S. Bloch and A. Ogus [BO]. This reformulation enables us to clarify and unify our previous definition ..."
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In this paper, we reformulate “morphic cohomology ” as introduced by the author and H.B. Lawson [FL1] in such a way that it and “Lawson homology ” satisfy the list of basic properties codified by S. Bloch and A. Ogus [BO]. This reformulation enables us to clarify and unify our previous definitions and provides this topological cycle theory with foundational properties which have proved useful for other cohomology theories. One formal consequence of these BlochOgus properties is the existence of a localtoglobal spectral sequence which should prove valuable for computations (as shown in Corollary 7.2). The basic result of this paper is that topological cycle cohomology theory (which agrees with morphic cohomology for smooth varieties) in conjunction with topological cycle homology theory (which is shown to always agree with Lawson homology) do indeed satisfy the BlochOgus properties for a “Poincaré duality theory with supports ” on complex quasiprojective varieties. We view the challenge of verification of the BlochOgus properties as worthy for several reasons. First, the properties require certain definitions and constructions
THE SYNTOMIC REGULATOR FOR K–THEORY OF FIELDS
"... We define complexes analogous to Goncharov’s complexes for the K–theory of discrete valuation rings of characteristic zero. Under suitable assumptions in K–theory, there is a map from the cohomology of those complexes to the K–theory of the ring. In case the ring is the localization of the ring of ..."
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We define complexes analogous to Goncharov’s complexes for the K–theory of discrete valuation rings of characteristic zero. Under suitable assumptions in K–theory, there is a map from the cohomology of those complexes to the K–theory of the ring. In case the ring is the localization of the ring of integers in a number field, there are no assumptions necessary. We compute the composition of our map to the K–theory with the syntomic regulator. The result can be described in terms of a p–adic polylogarithm. Finally, we apply our theory in order to compute the regulator to syntomic cohomology on Beilinson’s cyclotomic elements. The result is again given by the p–adic polylogarithm. This last result is related to one by Somekawa and generalizes work by Gros.
Limits in ncategories
, 1997
"... One of the main notions in category theory is the notion of limit. Similarly, one of the most commonly used techniques in homotopy theory is the notion of “homotopy limit” commonly called “holim ” for short. The purpose of the this paper is to begin to develop ..."
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One of the main notions in category theory is the notion of limit. Similarly, one of the most commonly used techniques in homotopy theory is the notion of “homotopy limit” commonly called “holim ” for short. The purpose of the this paper is to begin to develop
KTHEORY OF SEMILOCAL RINGS WITH FINITE COEFFICIENTS AND ÉTALE COHOMOLOGY
"... 3. Proofs of theorem 1, theorem 2 and corollary 1 10 4. Higher Chern classes with values in truncated étale cohomology 18 ..."
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3. Proofs of theorem 1, theorem 2 and corollary 1 10 4. Higher Chern classes with values in truncated étale cohomology 18
Flexible sheaves
, 1996
"... This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector b ..."
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This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector bundles, preprint of Toulouse 3, and also available at my homepage 2) for a more detailed introduction, and also for a further development of the ideas presented here. The present paper was finished in December 1993 while I was visiting MIT. Since writing this, I have realized that the theory sketched here is essentially equivalent to JardineIllusie’s theory of presheaves of topological spaces (although they talk about presheaves of simplicial sets which is the same thing). This is the point of view adopted in “Homotopy over the complex numbers and generalized de Rham cohomology”. 1 Added in August 1996: I am finally sending this to “Duke eprints ” because it now seems that there may be several useful features of the treatment given here. First of all, the direct construction of the homotopysheafification (by doing a certain operation n+2 times) seems to be useful, for example I need
Secondary KodairaSpencer classes and nonabelian Dolbeault cohomology
, 1998
"... One of the nicest things about variations of Hodge structure is the “infinitesimal variation of Hodge structure ” point of view [25]. A variation of Hodge structure (V = V p,q, ∇) over a base S gives rise at any point s ∈ S to the KodairaSpencer map κs: T(S)s → Hom(V p,q s, V p−1,q+1 s). In the geo ..."
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One of the nicest things about variations of Hodge structure is the “infinitesimal variation of Hodge structure ” point of view [25]. A variation of Hodge structure (V = V p,q, ∇) over a base S gives rise at any point s ∈ S to the KodairaSpencer map κs: T(S)s → Hom(V p,q s, V p−1,q+1 s). In the geometric situation of a family X → S we have V p,q s = Hq (Xs, Ω p
Hodge filtered complex bordism
, 2012
"... Abstract. We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex a ..."
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Abstract. We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle formula and A 1homotopy invariance. Moreover, we obtain transfer maps along projective morphisms. 1.
SHEAVES AND HOMOTOPY THEORY
"... The purpose of this note is to describe the homotopytheoretic version of sheaf theory developed in the work of Thomason [14] and Jardine [7, 8, 9]; a few enhancements are provided here and there, but the bulk of the material should be credited to them. Their work is the foundation from which Morel ..."
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The purpose of this note is to describe the homotopytheoretic version of sheaf theory developed in the work of Thomason [14] and Jardine [7, 8, 9]; a few enhancements are provided here and there, but the bulk of the material should be credited to them. Their work is the foundation from which Morel and Voevodsky build their homotopy theory for schemes [12], and it is our hope that this exposition will be useful to those striving to understand that material. Our motivating examples will center on these applications to algebraic geometry. Some history: The machinery in question was invented by Thomason as the main tool in his proof of the LichtenbaumQuillen conjecture for Bottperiodic algebraic Ktheory. He termed his constructions ‘hypercohomology spectra’, and a detailed examination of their basic properties can be found in the first section of [14]. Jardine later showed how these ideas can be elegantly rephrased in terms of model categories (cf. [8], [9]). In this setting the hypercohomology construction is just a certain fibrant replacement functor. His papers convincingly demonstrate how many questions concerning algebraic Ktheory or étale homotopy theory can be most naturally understood using the model category language. In this paper we set ourselves the specific task of developing some kind of homotopy theory for schemes. The hope is to demonstrate how Thomason’s and Jardine’s machinery can be built, stepbystep, so that it is precisely what is needed to solve the problems we encounter. The papers mentioned above all assume a familiarity with Grothendieck topologies and sheaf theory, and proceed to develop the homotopytheoretic situation as a generalization of the classical case. In some sense the approach here will be the reverse of this: we will instead assume a general familiarity with homotopy theory, and show how the theory of sheaves fits in with perspectives already offered by the field.