Intersection de sous-groupes et de sous-variétés I

We study the intersection of a subvariety X of an abelian variety A over with the union of all the algebraic subgroups of A of given dimension d. Our main result states that if we remove a suitable exceptional subset from X and if d is small enough then the intersection enjoys a Northcott-like property: the points of bounded height on it form a finite set. The condition on d involves only the dimension of X and the structure of A up to isogenies. We show how it can be weakened if we assume certain conjectures in the direction of an abelian version of Lehmer's problem. The theorem is especially meaningful when X is a curve since it is then possible to bound the height and hence to prove finiteness of the set under consideration. This generalises the result of E. Viada on powers of elliptic curves and is analogous to work of E. Bombieri, D. Masser and U. Zannier on tori, whose general strategy we follow.     

Inégalités de Brascamp-Lieb et convexité

This note gives an elementary proof of an inequality of Brascamp-Lieb and of a new dual inequality that has numerous applications to convexity: lower estimates of volumes of projections, of exterior volume ratio and MM^* -estimate in the non symmetric case.     

Lemmes de multiplicités et constante de Seshadri

Résumé  On démontre dans cet article un raffinement des lemmes de multiplcités de Philippon, essentiellement dans le cas particulier où l’on dérive dans toutes les directions. L’amélioration est rendue possible grace à un point de vue plus géométrique et notamment par l’apparition nouvelle dans ce contexte de la notion de constante de Seshadri.     

Remplissage et surfaces de révolution

The filling function of a Riemannian manifold is either linear or at least quadratic. This fact was originally discovered by M. Gromov in 1985. We address the question of the existence of further obstructions. We give a partial answer: every superadditive and superquadratic function is asymptotic to the filling function of a surface of revolution. A function which furthermore satisfies a natural second-order differential inequation is equal to a filling function.     

Caractérisation et existence de structures de Dirac multiplicatives

We define the product of two Dirac manifolds and introduce the notion of a Dirac–Lie group of Poisson type. This notion is equivalent to that of multiplicative Dirac structure and any real simply-connected Lie group carries a no trivial multiplicative Dirac structure when its dimension is at least 2. To cite this article: A. Affane, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg

In this paper, we prove dispersive and Strichartz inequalities on the Heisenberg group. The proof involves the analysis of Besov-type spaces on the Heisenberg group.

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Le travail du troisième auteur est partiellement finacé par la NNSF de Chine.     

Résolution de certains modules instables et fonction de partition de Minc

One constructs minimal injective resolutions for certain unstable modules that appear to be the mod 2 cohomology of Thom spectra. The terms of the resolution are tensor products of Brown–Gitler modules and Steinberg modules introduced by S. Mitchell and S. Priddy. A combinatorial result of Andrews shows that the alternating sum of the Poincaré series of the considered modules is zero. One gives homotopical applications of this result. To cite this article: D.H.H. Nguyen et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Groupoïde d'holonomie et géométrie globale

The purpose of this Note is to show that global geometry requires Lie groupoid theory. In this way we show that holonomy groupoid of a regular foliation leads to a satisfactory solution of many global existence problems. Three examples are given: Palais'theorem on local group actions, a characteristics method and links between Lie groupoid and algebroid morphisms. Complete proofs will be given elsewere.     

3-Instantons et réseaux de quadriques

Sans résumé     

Filtration perverse et théorie de Hodge

The compatibility of the perverse filtration with Hodge theory with coefficients in an admissible variation of a mixed Hodge structure on the complement of a normally crossing divisor is established using a logarithmic complex, with a view to obtaining a new proof of the decomposition theorem.     

Mauvaise réduction des variétés de Drinfeld et correspondance de Langlands locale

Sans résumé Oblatum 28-XII-1998 & 3-V-1999 / Published online: 5 August 1999