Réduction de la conjecture de Hodge à une continuité

The obstructions, for a closed differential form on a projective manifold, to be cohomologous to an algebraic cycle with complex coefficients, are computed in terms of the Chow transformation. They can be expressed as an orthogonality condition, on the manifold itself, with families parametrized by the Grassmannian of currents which are completely determined. A parameter does not yield any obstruction if the associated projective subspace meets properly the manifold. The embedding of the manifold is degenerated, in view of applying the characterization of currents associated to algebraic cycles by the Chow transformation. We study the set of periods obtained when the parameter varies, in particular, we prove a continuity result, thanks to the constructibility of the Bernstein polynomial. When the cohomology class is rational, we conjecture that this set is connected.     

Sur un problème de potentiel avec conditions aux limites mixtes: une conjecture de A. Acker et sa démonstration à l'aide de problèmes auxiliaries à une dimension

Résumé On étudie quelques problèmes relatifs aux fonctions harmoniques satisfaisant des conditions aux limites de type radiation, par des méthodes de variation. On démontre notamment une conjecture de A. Acker [1, 2] à l'aide de problèmes auxiliaires à une dimension.

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Summary Some problems for harmonic functions with radiation-type boundary conditions are studied, and a conjecture of A. Acker [1, 2] is proved by using one-dimensional auxiliary problems.

     

Décompte dans une conjecture de Lang

Faltings has proven the following conjecture of Lang: if A is an abelian variety over a number field and X any subvariety then all rational points of X lie on a finite number N of translates, contained in X, of abelian subvarieties of A. We provide an upper bound for N whose main feature is uniformity in X since it does not depend on the height of X. Moreover, the bound is completely explicit. Together with the result of a previous paper, the heart of the proof is a suitable generalization of Mumford’s theorem for curves.

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Décompte dans une conjecture de Lang

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Oblatum 20-III-2000 & 3-V-2000?Published online: 16 August 2000     

Sur une note de Nagata relative à un problème de Krull

Sans résumé     

Triangles inscrits et circonscrits à une courbe convexe sphérique

Sans résumé Présenté par G. Hajós     

La conjecture de Manin g��om��trique pour une famille de quadriques intrins��ques

We prove a version of Manin??s conjecture for a certain family of intrinsic quadrics, the base field being a global field of positive characteristic. We also explain how a very slight variation of the method we use allows to establish the conjecture for a certain generalized del Pezzo surface.     

Convergence des processus de Nelson-Aalen et de Kaplan-Meier par une méthode de martingale

A new martingale method for proving weak convergence results, which allows discontinuities for the underlying distribution, is introduced and applied to the empirical processes of Nelson-Aalen and Kaplan-Meier.     

Sections hyperplanes à singularités simples et exemples de variations de structure de Hodge

On construit des variétés projectives lisses complexes de dimension 3 à 6 munies de variations de structure de Hodge, en généralisant une construction que J. Carlson et C. Simpson avaient effectuée en dimension 2. Ensuite, on étudie quelques unes de leurs propriétés, en particulier leurs groupes de cohomologie.     

Régularité dans une équation de Schrödinger avec potentiel singulier à distance finie et à l'infini

In this Note we study the Schrödinger equation i?_tu+Δu+V₀u+V₁u=0 on $\text{R}{}^3\text{×(0,T)}$ with initial condition $\text{u}{}_0\text{∈{v∈H}{}^2\text{(}\text{R}{}^3\text{)}$, where V₀ is a coulombian potential, singular at finite distance and V₁ is an electric potential, possibly unbounded. Both of them may depend on space and time variables. We prove that this problem is well-posed and that the regularity of the initial data is conserved for the solution. The detailed proof will be given elsewhere (Baudouin et al., in press). To cite this article: L. Baudouin et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Un théorème de Torelli infinitésimal à coefficients

We generalize an infinitesimal Torelli theorem for hypersurfaces of high degree of M. Green (1985) [1], for twisted coefficients.     

Sur les égalités du miroir et certaines formes faibles de la conjecture de Greenberg

Let p be an odd prime. For any CM number field K containing a primitive p^th-root of unity, class field theory and Kummer theory put together yield the well known reflection inequality ^+– between the plus and minus parts of the -invariant of K. Greenbergs conjecture asserts that ⁺ is always trivial. We study here a weak form of this conjecture, namely ⁺=^– if and only if ⁺=^–=0.     

Sur la courbure d’une série de surfaces et de lignes

Annali di Matematica Pura ed Applicata (1858-1865) -     

Sur la division de formes et de courants par une forme linéaire

Sans résumé Dédié à Monsieur Heinz Hopf pour son soixantième anniversaire     

Un problème de confinement pour une coque membranaire linéairement élastique de type elliptique

We identify, and rigorously justify by an asymptotic analysis, the variational inequalities of the two-dimensional problem satisfied by the displacement field of a linearly elastic membrane shell of elliptic type subjected to a confinement condition inside a half-space. This type of condition substantially differs from the Signorini condition usually imposed on the “lower face” of the shell.     

Modèles de verre de spins à champ moyen et à interactions multiples : résultats rigoureux à basse température

We study the Ising case of the p-spin interaction mean field model for spin glasses. For large p, below the critical temperature, the configuration space naturally decomposes in a sequence of “states”. In the thermodynamic limit, the overlaps between two generic elements of any two different “states” are zero.We also construct systems that provably exhibit several phase transitions between what one might in replica theory call different levels of symmetry breaking.