Algèbre de Clifford symplectique. Revêtements du groupe symplectique. Indices de Maslov et spineurs symplectiques

Sans résumé

---

This article is the developed text of a talk given at the Symposium on Differential Geometry in Debrecen, Hungary, on August 28–September 3, 1975.     

Cocycles d'Euler et de Maslov

Conclusion Nous espérons avoir convaincu le lecteur qu'il peut être utile de considérer la classe de Maslov comme une classe bornée. Dans [Gh], nous avons montré que la classe d'Euler bornée pour un groupe d'homéomorphismes directs du cercle rend compte de la dynamique topologique de ce groupe. Existe-t-il un résultat analogue pour Sp(2n,)? En d'autres termes, soit un groupe discret et ₁, ₂ deux représentations de dans Sp(2n,). On suppose que les cocycles ₁ ^* et ₂ ^* définissent la même classe bornée. Que peut-on en conclure sur ₁ et ₂?Par ailleurs, l'article [At l] traite aussi d'invariants sur SL(2,) différents de ceux que nous avons considérés, comme par exemple les fonctionsL de Shimizu. Est-il possible de les faire rentrer naturellement dans notre cadre?

---

     

Cellules quantiques symplectiques et fonctions de Husimi-Wigner

We propose a definition of quantum cells which is invariant under symplectic transformations. We use this notion to the study of positivity properties of the Wigner and Husimi functions, which allows us to precise and to improve known results.     

Courants positifs extrêmaux et conjecture de Hodge

Sans résumé     

Obstruction à la transversalité de lagrangiens

In this Note, we give a necessary and sufficient condition for Lagrangians in a symplectic vector bundle to be deformed stably into transversal Lagrangians. In the case of three Lagrangians, we show that the associated Grothendieck group can be identified with a Hermitian K-theory group. To cite this article: M. Karoubi, M.L. Lapa de Souza, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Prolongement de courants positifs a travers de petits obstacles

In this paper, we prove an extension theorem through closed subsets having small Haussdorff dimension, for positive currents whose boundary satisfies some growth condition. As a corollary, we get the classical Harvey's extension theorem for closed positive currents. Furthermore, we apply our result to study the boundary of holomorphic chains.

     

L''interpretation de la classe de Maslov dans la K-theorie Hermitienne et dans la theorie relative de Chern-Weil

The purpose of this paper is, first to define the Maslov invariant in the U-theory, the intermediate theory between real K-theory and complex K-theory, whose vanishing is a necessary and sufficient condition for the stable transversality of two Lagrangian sub-bundles of a symplectic fiber bundle. The second purpose is to show, in conformity with Bott periodicity, that Chern, Pontryand Stiefel-Whitney classes are precise enough to play the same role as the Maslov classes when one replaces the base space X of the symplectic bundle, by S ⁷ X ⁺the seventh topological suspension of X ⁺the compactification of X.     

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.     

Classification locale simultanée de deux formes symplectiques compatibles

Consider a symplectic form ω and a closed 2-form ω₁ on a real or complex manifold. Suppose that the Nijenhuis torsion of the tensor fieldJ defined by ω₁(X,Y) = ω(JX,Y) vanishes. In this paper we give the complete local classification of the couple {ω, ω₁} on a dense open set, defined by some minor conditions of regularity. Around each point of this open set we can find coordinates on wich ω is written with constant coefficients and ω₁ with affine ones. Projet de recherche DGICYT PB91-0412     

Tranchage et prolongement des courants positifs fermés

Sans résumé Re?u le 11 octobre 1995 / Version revisée re?ue le 20 fevrier 1996     

Sur les tissus plans de rang maximal et le problème de Chern

We explain how it is possible to obtain the necessary conditions on functions defining a germ of planar web in order that it be of maximal rank. Then we apply this method to the study of maximal rank 5-webs. We show the existence of many exceptional 5-webs non-equivalent to Bol's web, thus giving an answer to Chern's problem. To cite this article: L. Pirio, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Les Plans de Benz: Une approche unifiee des plans de Moebius,Laguerre et Minkowski

Sans résuméArticle dédié au Professeur Raphael Artzy, à l' occasion de. son 70^e anniversaire.     

Actions symplectiques de groupes compacts (Symplectic Actions of Compact Groups)

For any symplectic action of a compact connected group on a compact connected symplectic manifold, we show that the intersection of the Weyl chamber with the image of the moment map is a closed convex polyhedron. This extends Atiyah–Guillemin–Sternberg–Kirwan's convexity theorems to non-Hamiltonian actions. As a consequence, we describe those symplectic actions of a torus which are coisotropic (or multiplicity free), i.e. which have at least one coisotropic orbit: they are the product of an Hamiltonian coisotropic action by an anhamiltonian one. The Hamiltonian coisotropic actions have already been described by Delzant thanks to the convex polyhedron. The anhamiltonian coisotropic actions are actions of a central torus on a symplectic nilmanifold. This text is written as an introduction to the theory of symplectic actions of compact groups since complete proofs of the preliminary classical results are given. An erratum to this article is available at .     

Construction de plans par remplacement de secteurs

Sans résumé     

Prolongement des courants,positifs, fermés de masse finie

Sans résumé     

Compacité des opérateurs de Dunford–Pettis positifs sur les treillis de Banach

We establish some sufficient and necessary conditions which give the compactness of positive Dunford–Pettis operators on Banach lattices. To cite this article: B. Aqzzouz et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Sur les opérateurs de Dunford-Pettis positifs qui sont faiblement compacts

Nous donnons des conditions nécessaires et suffisantes pour que tout opérateur de Dunford-Pettis positif sur un treillis de Banach, soit faiblement compact et nous déduisons quelques conséquences.