Caractère de Chern bivariant

We construct a bivariant Chern character with values in Jones-Kassel's bivariant cyclic cohomology. This is done forK-theoretic objects such as idempotents, bimodules, quasi-homomorphisms à la Cuntz and extensions of algebras.

     

Classes polaires associées aux distributions holomorphes de sous-espaces tangents*

Résumé. Nous définissons des classes polaires associées à une distribution holomorphe singulière de sous-espaces tangents d’une variété projective lisse. Nous prouvons que ces classes polaires peuvent être calculées en fonction des classes de Chern-Mather du faisceau tangent de la distribution et réciproquement. Nous utilisons leurs degrés pour borner les degrés de certaines classes polaires associés à une variété invariante. *Ce travail a été soutenu par l’Accord de Coopération France-Brésil (CNRS/CNPq) et par la Fondation CAPES-Brésil.     

Equivariant–Bivariant Chern Character for Profinite Groups

For the action of a locally compact and totally disconnected group G on a pair of locally compact spaces X and Y we construct, by sheaf theoretic means, a new equivariant and bivariant cohomology theory. If we take for the first space Y an universal proper G-action then we obtain for the second space its delocalized equivariant homology. This is in exact formal analogy to the definition of equivariant K-homology by Baum, Connes, Higson starting from the bivariant equivariant Kasparov KK-theory. Under certain basic finiteness conditions on the first space Y we conjecture the existence of a Chern character from the equivariant Kasparov KK-theory of Y and X into our cohomology theory made two-periodic which becomes an isomorphism upon tensoring the KK-theory with the complex numbers. This conjecture is proved for profinite groups G. An essential role in our construction is played by a bivariant version of Segal localization which we establish for KK-theory.     

Classes Caractéristiques de Fibrés Feuilletés, Holomorphes ou Algébriques

Résumé L'objet de cet article est de montrer comment laK-théorie multiplicative peut être utilisée pour définir de nouvelles classes caractéristiques secondaires de fibrés vectoriels munis de structures supplémentaires, dans des contextes géométriques variés. On y développe des exemples reliés à des travaux antérieurs divers.

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The purpose of this paper is to show how multiplicativeK-theory may be used to define new secondary characteristic classes for vector bundles with additional structures, in various geometrical contexts. Many examples are developed which are related to previous works.

     

Chern character for twisted K-theory of orbifolds

For an orbifold X and α∈H³(X,Z), we introduce the twisted cohomology and prove that the non-commutative Chern character of Connes-Karoubi establishes an isomorphism between the twisted K-groups and the twisted cohomology . This theorem, on the one hand, generalizes a classical result of Baum-Connes, Brylinski-Nistor, and others, that if X is an orbifold then the Chern character establishes an isomorphism between the K-groups of X tensored with C, and the compactly-supported cohomology of the inertia orbifold. On the other hand, it also generalizes a recent result of Adem-Ruan regarding the Chern character isomorphism of twisted orbifold K-theory when the orbifold is a global quotient by a finite group and the twist is a special torsion class, as well as Mathai-Stevenson's theorem regarding the Chern character isomorphism of twisted K-theory of a compact manifold.     

Invariants de Maslov équivariants

We construct two cohomological invariants associated to pairs of Lagrangian sub-bundles of a symplectic bundle on a compact manifold upon which a compact Lie group is acting. The first invariant, which we call the classical equivariant Maslov H-invariant, provides an obstruction to Lagrangian transversality and lives in the Borel cohomology. The second invariant, which we call the equivariant Maslov U-invariant, generalises the author's results in K-Theory 13 (1998), 347–361 to the equivariant context and provides a necessary and sufficient condition for equivariant Lagrangian transversality, up to homotopic stability, and lives in the U-theory (intermediate between the real complex K-theories). As an application, we show that two Lagrangian sub-bundles of a symplectic bundle on a homogeneous space are always stably transverse.     

K-Theory of Semi-local Rings with Finite Coefficients and étale Cohomology

Let A be a commutative semi-local ring containing 1/2. We construct natural isomor-phisms

Isomorphism Classes of A-Hypergeometric Systems

Given a finite set A of integral vectors and a parameter vector, Gel'fand, Kapranov, and Zelevinskii defined a system of differential equations, called an A-hypergeometric (or a GKZ hypergeometric) system. Classifying the parameters according to the D-isomorphism classes of their corresponding A-hypergeometric systems is one of the most fundamental problems in the theory. In this paper we give a combinatorial answer for the problem under the assumption that the finite set A lies in a hyperplane off the origin, and illustrate it in two particularly simple cases: the normal case and the monomial curve case.     

Milnor K-Groups and Finite Field Extensions

Let E/F be a finite separable field extension and let m denote the integral part of log₂ [E : F]. David Leep recently showed that if char(F) 2, then for n m the nth power of the fundamental ideal in the Witt ring of E satisfies the equality I ⁿ E = I ^n–m F · I ^m E. The aim of this note is to prove the analogous equality for the Milnor K-groups, that is K _n E = K _n–m F · K _m E for n m. In either of these equalities one may not replace m by m – 1, as examples of certain m-quadratic extensions indicate.     

On Milnor K-groups attached to semi-Abelian varieties

In this paper we define and study a Milnor K-group attached to a finite family of semi-Abelian varieties over a field, which is a generalization of the usual Milnor K-group. Using this group, we generalize the work of Bloch.     

Terme constant de fonctions sur un espace symétrique réductif p-adique

We generalize Casselman's pairing to p-adic reductive symmetric spaces and study the asymptotic behaviour of certain generalized coefficients. We also prove an analogue of a lemma due to Langlands which allows us to prove a disjunction result for the Cartan decomposition of the p-adic reductive symmetric spaces.     

Approximation de compacts fonctionnels par des variétés analytiques

In the theory of approximation there are some problems on approximation of compact sets in functional spaces by analytic families. First, we deal with the case of algebraic varieties, the theorem of Vitushkin, in which we give a new proof based on the method of Warren, with precision of constants. Next, we consider the case of analytic varieties which is as well a negative result: we show that an analytic family with N variables cannot approach the compact Λl,s better than order as N increases. We finish by giving some applications in Sturm-Liouville inverse theory.     

The Higher K-Theory of Complex Varieties

Let X be a smooth complex variety, and let F be its function field. We prove that (after localizing at the prime 2) the K-groups of F are divisible above the dimension of X, and that the K-groups of X are divisible-by-finite. We also describe the torsion in the K-groups of F and X.     

A Regulator Formula for Milnor K-groups

The classical Abel–Jacobi map is used to geometrically motivate the construction of regulator maps from Milnor K-groups K _n ^M (C(X)) to Deligne cohomology. These maps are given in terms of some new, explicit (n – 1)-currents, higher residues of which are defined and related to polylogarithms. We study their behavior in families X _s and prove a rigidity result for the regulator image of the Tame kernel, which leads to a vanishing theorem for very general complete intersections.     

Groups acting on buildings, operator K-theory, and Novikov''s conjecture

The paper is devoted to the study of the KK-theory of Bruhat-Tits buildings. We develop a theory which is analogous to the corresponding theory for manifolds of nonpositive sectional curvature. We construct a C ^*-algebra and a Dirac element associated to any simplicial complex. In the case of buildings, we construct, moreover, a dual Dirac element and compute its KK-products with the Dirac element. As a consequence, we prove the Novikov conjecture for discrete subgroups of linear adelic groups. In our study, we develop a KK-theoretic Poincaré duality for non-Hausdorff manifolds.Dedicated to Alexander Grothendieck on his sixtieth birthday     

Estimation de Yule–Walker d'un CAR(p) observé à temps discret

Let (X(lδ), l=0,n) be a discrete observation at mesh δ>0 of X, a CAR(p). Classical Yule–Walker estimation are biased and must be corrected. Resultant estimators converge if T=nδ→+∞, are asymptotically normal with rate , and efficient. The diffusion coefficient is also estimated, with rate .     

Une étude mathématique d'un modèle de Rotenberg

In this paper, a cell proliferation model of Rotenberg is mathematically studied. Each cell of this population is distinguished by its degree of maturity μ and its maturation velocity v. We equipped this mathematical model by boundary conditions wich generalize all biological rules imposed by Rotenberg. We show that this model is governed by a strongly continuous semigroup. We also develop some properties of this semigroup and we study its asymptotic behaviour in the uniform topology.