La 1-forme de torsion d'une variété hermitienne compacte

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Inégalités de hardy sur les variétés riemanniennes non-compactes

We give a criterium and its applications in order that a riemannian manifold satisfy a inequality similar to Hardy's inequality on Rⁿ.     

Formes linéaires de logarithmes effectives sur les variétés abéliennes

We prove new measures of linear independence of logarithms on an abelian variety defined over , which are totally explicit in function of the invariants of the abelian variety (dimension, Faltings height, degree of a polarization). Besides, except an extra-hypothesis on the algebraic point considered and a weaker numerical constant, we improve on earlier results (in particular David's lower bound). We also introduce into the main theorem an algebraic subgroup that leads to a great variety of different lower bounds. An important feature of the proof is the implementation of the slope method of Bost and some results of Arakelov geometry naturally associated with it.     

Approximation diophantienne sur les variétés semi-abéliennes

Let A be a semi-abelian variety over , Γ a subgroup of of finite rank and X a subvariety of A which is not a translate of a semi-abelian subvariety of A. Work by P. Vojta and M. McQuillan shows that is not dense in X. B. Poonen has then conjectured that the same remains true if Γ is replaced by a fattening for a certain ε>0 where h is a canonical height. B. Poonen and S. Zhang have shown independently this to hold when A is almost split. On the other hand, the statement contains the Bogomolov property (with Γ=0) now proven by S. David and P. Philippon. In this paper, we prove Poonen's conjecture for any A. We also consider the slightly more general sets instead of Γ_ε. We use the case Γ=0 as well as a generalized Vojta inequality.     

Cohomologie des variétés de Deligne-Lusztig

We study the cohomology of Deligne-Lusztig varieties with aim the construction of actions of Hecke algebras on such cohomologies, as predicted by the conjectures of Broué, Malle and Michel ultimately aimed at providing an explicit version of the abelian defect conjecture. We develop the theory for varieties associated to elements of the braid monoid and partial compactifications of them. We are able to compute the cohomology of varieties associated to (possibly twisted) rank 2 groups and powers of the longest element w₀ (some indeterminacies remain for G₂). We use this to construct Hecke algebra actions on the cohomology of varieties associated to w₀ or its square, for groups of arbitrary rank. In the subsequent work [F. Digne, J. Michel, Endomorphisms of Deligne-Lusztig varieties, Nagoya J. Math. 183 (2006)], we construct actions associated to more general regular elements and we study their traces on cohomology.     

Bornes sur les degrés dynamiques d'automorphismes de variétés kählériennes de dimension 3

I study the dynamical degrees of automorphisms of a compact Kähler manifold X   of dimension 3 and, more generally, the type of growth of the norm of (fⁿ)^?${(f^n)}^{?}$, where f^?$f^{?}$ denotes the action of the automorphism f on the cohomology of X. The automorphisms of complex tori show that the results are optimal.     

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.     

Forme hermitienne canonique sur la cohomologie de la fibre de Milnor d'une hypersurface à singularité isolée

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Borne sur la torsion dans les variétés abéliennes de type CM

Let A be an abelian variety of dimension g1 defined over a number field K. We study the size of the torsion group A(F)_tors where F/K is a finite extension and more precisely we study the best possible exponent γ in the inequality Card(A(F)_tors)[F:K]^γ when F is any finite extension of K. In the CM case we give an exact formula for the exponent γ in terms of the characters of the Mumford–Tate group—a torus in this case—and discuss briefly the general case.Finally we give an application of the main result in direction of a generalisation of the Manin–Mumford conjecture.     

Une remarque sur la symétrie asymptotique de la fonction de Green

Dans cette note, nous montrons que la symétrie asymptotique de la fonction de Green dans le cas des variétés compactes de courbure négative implique que la variété est localement symétrique.