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1.
Techniques,computations, and conjectures for semi-topological <Emphasis Type="Italic">K</Emphasis>-theory 总被引:1,自引:0,他引:1
We establish the existence of an Atiyah-Hirzebruch-like spectral sequence relating the morphic cohomology groups of a smooth, quasi-projective complex variety to its semi-topological K-groups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that relates the motivic cohomology and algebraic K-theory of varieties, and it is also compatible with the classical Atiyah-Hirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the Borel-Moore (singular) homology of complex varieties introduced by H. Gillet and C. Soulé – to compute the semi-topological K-theory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational three-folds, and related varieties, the semi-topological K-groups and topological K-groups are isomorphic in all degrees permitted by cohomological considerations. We also formulate integral conjectures relating semi-topological K-theory to topological K-theory analogous to more familiar conjectures (namely, the Quillen-Lichtenbaum and Beilinson-Lichtenbaum Conjectures) concerning mod-n algebraic K-theory and motivic cohomology. In particular, we prove a local vanishing result for morphic cohomology which enables us to formulate precisely a conjectural identification of morphic cohomology by A. Suslin. Our computations verify that these conjectures hold for the list of varieties above.Mathematics Subject Classification (2000): 19E20, 19E15, 14F43The first author was partially supported by the NSF and the NSAThe second author was supported by the Helen M. Galvin Fellowship of Northwestern UniversityThe third author was partially supported by the NSF and the NSA 相似文献
2.
Thomas Brélivet 《Bulletin des Sciences Mathématiques》2002,126(4):333-342
Using the theory of the mixed Hodge structure one can define a notion of spectrum of a singularity or of a polynomial. Recently Claus Hertling proposed a conjecture about the variance of the spectrum of a singularity. Alexandru Dimca proposed a similar conjecture on polynomials. Here, we prove these two conjectures in the case of dimension 2 and when the singularity or the polynomial is Newton non-degenerated and commode. 相似文献
3.
4.
Uwe Jannsen 《Japanese Journal of Mathematics》2010,5(1):73-102
The concept of weights on the cohomology of algebraic varieties was initiated by fundamental ideas and work of A. Grothendieck
and P. Deligne. It is deeply connected with the concept of motives and appeared first on the singular cohomology as the weights
of (possibly mixed) Hodge structures and on the etale cohomology as the weights of eigenvalues of Frobenius. But weights also
appear on algebraic fundamental groups and in p-adic Hodge theory, where they become only visible after applying the comparison functors of Fontaine. After rehearsing various
versions of weights, we explain some more recent applications of weights, e.g. to Hasse principles and the computation of
motivic cohomology, and discuss some open questions. 相似文献
5.
Elham Izadi 《Annali di Matematica Pura ed Applicata》2010,189(3):487-495
We show how the classical Hodge conjecture for the middle cohomology of an abelian variety is equivalent to the general Hodge
conjecture for the middle cohomology of a smooth ample divisor in the abelian variety. This is best suited to abelian varieties
with actions of imaginary quadratic fields. 相似文献
6.
Adrian Vasiu 《Mathematische Nachrichten》2020,293(12):2399-2448
We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic (0, p). As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic (0, p) of integral canonical models of projective Shimura varieties of Hodge type with respect to h-hyperspecial subgroups as pro-étale covers of Néron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle. 相似文献
7.
We prove in full generality the mirror duality conjecture for string-theoretic Hodge numbers of Calabi–Yau complete intersections
in Gorenstein toric Fano varieties. The proof is based on properties of intersection cohomology.
Oblatum 9-X-1995 & 11-III-1996 相似文献
8.
Claire Voisin 《Milan Journal of Mathematics》2010,78(1):85-116
These notes discuss Hodge theory in the algebraic and Kähler context. We introduce the notion of (polarized) Hodge structure on a cohomology algebra and show how to extract from it topological restrictions on compact Kähler manifolds, and stronger topological restrictions on projective complex manifolds. The second part of the notes is devoted to the discussion of the Hodge conjecture, showing in particular that there is no way to extend it to the Kähler context. We will also discuss algebraic de Rham cohomology which is specific to projective complex manifolds and allows to formulate a number of arithmetic questions related to the Hodge conjecture. 相似文献
9.
Laurent Manivel 《Inventiones Mathematicae》1997,127(2):401-416
Summary. The main result of this article is a general vanishing theorem for the cohomology of tensorial representations of an ample
vector bundle on a smooth complex projective variety. In particular, we extend classical theorems of Griffiths and Le Potier
to the whole Dolbeault cohomology, prove a variant of an uncorrect conjecture of Sommese, and answer a question of Demailly.
As an application, we prove conjectures of Debarre and Kim for branched coverings of grassmannians, and extend a well-known
Barth–Lefschetz type theorem for branched covers of projective spaces, due to Lazarsfeld. We also obtain new restriction theorems
for certain degeneracy loci.
Oblatum 10-IV-1996 & 22-V-1996 相似文献
10.
We prove a formula expressing the motivic integral (Loeser and Sebag, 2003) [34] of a K3 surface over C((t)) with semi-stable reduction in terms of the associated limit mixed Hodge structure. Secondly, for every smooth variety over a complete discrete valuation field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerate K3 surfaces over an arbitrary complete discrete valuation field and prove this conjecture for Kummer K3 surfaces. 相似文献
11.
B. Kahn 《Mathematische Annalen》2002,324(3):581-617
We reformulate part of the arguments of T. Geisser and M. Levine relating motivic cohomology with finite coefficients to
truncated étale cohomology with finite coefficients [9,10]. This reformulation amounts to a uniqueness theorem for motivic
cohomology, and shows that the Geisser-Levine method can be applied generally to compare motivic cohomology with other types
of cohomology theories. We apply this to prove an equivalence between conjectures of Tate and Beilinson on cycles in characteristic
p and a vanishing conjecture for continuous étale cohomology.
Received: 23 November 2000 / Published online: 5 September 2002 相似文献
12.
Guillermo Cortiñas 《K-Theory》2000,20(2):175-200
We study a noncommutative version of the infinitesimal site of Grothendieck. A theorem of Grothendieck establishes that the cohomology of the structure sheaf on the infinitesimal topology of a scheme of characteristic zero is de Rham cohomology. We prove that, for the noncommutative infinitesimal topology of an associative algebra over a field of characteristic zero, the cohomology of the structure sheaf modulo commutators is periodic cyclic cohomology. We also compute the noncommutative infinitesimal cohomology of other sheaves. For example, we show that infinitesimal hypercohomology with coefficients in K-theory gives the fiber of the Jones–Goodwillie character which goes from K-theory to negative cyclic homology. 相似文献
13.
Claire Voisin 《Geometric And Functional Analysis》2010,19(5):1494-1513
Griffiths computation of the Hodge filtration on the cohomology of a smooth hypersurface X of degree d in
\mathbbPn{\mathbb{P}^n} shows that it has coniveau ≥ c once n ≥ dc. The generalized Hodge conjecture (GHC) predicts that the cohomology of X is then supported on a closed algebraic subset of codimension at least c. This is essentially unknown for c ≥ 2. In the case where c = 2, we exhibit a geometric phenomenon in the variety of lines of X explaining the estimate for the coniveau, and show that (GHC) would be implied in this case by the following conjecture on
effective cones of cycles of intermediate dimension: Very moving subvarieties have their class in the interior of the effective
cone. 相似文献
14.
The aim of this paper is to prove the weight-monodromy conjecture (Delignes conjecture on the purity of monodromy filtration) for varieties p-adically uniformized by the Drinfeld upper half spaces of any dimension. The ingredients of the proof are to prove a special case of the Hodge standard conjecture, and apply a positivity argument of Steenbrink, M. Saito to the weight spectral sequence of Rapoport-Zink. As an application, by combining our results with the results of Schneider-Stuhler, we compute the local zeta functions of p-adically uniformized varieties in terms of representation theoretic invariants. We also consider a p-adic analogue by using the weight spectral sequence of Mokrane. 相似文献
15.
Seung Jin Lee 《Journal of Algebraic Combinatorics》2018,47(2):213-231
We discuss a relationship between Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds, the Fomin–Kirillov algebra, and the generalized nil-Hecke algebra. We show that the nonnegativity conjecture in the Fomin–Kirillov algebra implies the nonnegativity of the Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of Bott–Samelson varieties. We also discuss refined positivity conjectures of the Chern–Schwartz–MacPherson classes for Schubert cells motivated by the nonnegativity conjecture in the Fomin–Kirillov algebra. 相似文献
16.
Vincenzo Ancona 《Bulletin des Sciences Mathématiques》2006,130(6):525-552
In recent publications, we have defined complexes of differential forms on analytic spaces which are resolutions of the constant sheaf. These complexes were used to prove the existence of a mixed Hodge structure on the cohomology of analytic spaces which possess kählerian hypercoverings, in particular, projective algebraic varieties. We define an exterior product on these forms, which induces the cup product on the cohomology of analytic spaces. The main difficulty is to prove that this exterior product is functorial with respect to morphisms of analytic spaces. This exterior product can be used to prove that the cup product is compatible with the mixed Hodge structure on the cohomology. 相似文献
17.
Geordie Williamson 《Acta Mathematica》2016,217(2):341-404
We prove the local hard Lefschetz theorem and local Hodge–Riemann bilinear relations for Soergel bimodules. Using results of Soergel and Kübel, one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction. 相似文献
18.
In this paper we construct a bivariant version of cyclic cohomology and study its fundamental properties. We prove universal coefficient theorems relating the bivariant theory with cyclic homology and cohomology, we construct products in the bivariant theory, and we analyse the notion of an HC-equivalence.Dedicated to Alexander Grothendieck 相似文献
19.
Xin ZHANG 《Frontiers of Mathematics in China》2018,13(5):1189-1214
We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem. 相似文献
20.
We calculate the E-polynomials of certain twisted GL(n,ℂ)-character varieties of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example,
the value of the topological Euler characteristic of the associated PGL(n,ℂ)-character variety. The calculation also leads to several conjectures about the cohomology of : an explicit conjecture for its mixed Hodge polynomial; a conjectured curious hard Lefschetz theorem and a conjecture relating
the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n=2. 相似文献