Prolongement méromorphe des courants de Green

Résumé. Sur une variété quasi-projective complexe, on construit des courants dépendant d'un paramètre holomorphe qui prolongent les courants d'intégration des sous-variétés et les courants de Green de type logarithmique. On prouve des résultats de régularité et d'holomorphie pour ces courants et pour leur produits. On démontre que le *-produit dans la théorie d'Arakelov peut être défini par prolongement méromorphe à partir du produit de courants dépendant d'un paramètre dans une région de l'espace des paramètres où ils sont représentés par des formes différentielles. On donne une nouvelle preuve pour la commutativité et pour l'associativité du *-produit. Revised version: 16 July 2001 / Published online: 1 February 2002     

Prolongement des courants,positifs, fermés de masse finie

Sans résumé     

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.

     

Prolongement des espaces analytiques normaux

Sans résumé     

Singularités des courants d'Ahlfors

We prove that an algebraic curve charged by a current coming from an entire curve is rational or elliptic. This answers a question by M. P?un.     

Tranchage et prolongement des courants positifs fermés

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

Prolongement Unique Des Solutions

We prove a unique continuation property for solutions of Stokes equations with a non regular potential. For this, we state a Carleman's inequality which concerns the Laplace operator.     

Prolongement de faisceaux analytiques coherents

Sans résumé     

Estimation effective de la perte de positivité dans la régularisation des courants

Let (X,ω) be a compact complex Hermitian manifold, and let T?γ be a d-closed (1,1) almost positive current on X. A variant of Demailly's regularization-of-currents theorem states that T is the weak limit of a sequence of (1,1)-currents T_m with analytic singularities of coefficient 1/m, lying in the same cohomology class as T, whose Lelong numbers converge to those of T, and with a loss of positivity decaying to zero. We prove that if the (1,1)-form γ is assumed to be closed and C^∞, the regularizing currents T_m can be chosen such that $\text{T}{}_{\mathrm{m}}\text{?γ?}\text{C}\text{m}$ for a constant C>0 independent of m. To cite this article: D. Popovici, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Prolongement d'un courant positif plurisousharmonique

The purpose of this Note is to prove an extension result for a positive plurisubharmonic (Psh) current T defined in the complement of a closed complete pluripolar set A, under the hypothesis that $H_{2p}(\overline{\mathrm{supp}T}\cap A)=0$ and that $d{d}^{c}T$ has a locally finite mass. We prove in the first part an Oka type inequality for a positive Psh current. We then prove an extension result for such a current across an irreductible analytic set, thereby generalizing a result of Siu concerning positive closed current and one of our previous results for a negative Psh current. To cite this article: K. Dabbek, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Sur la laminarité de certains courants

We show that a sequence of smooth analytic curves of the unit ball of the complex plane, for which the genus is bounded by the area, converges to a lamination in a weak sense.     

Amplificateur param'etrique pour la mesure de faibles courants continus

Sans résumé     

Courants algébriques et courants de Liouville avec conditions sur les tranches

In this Note, we study closed positive currents with algebraic or Liouville slices (parallel or concurrent). We improve some results due to Blel, Mimouni and Raby in the case of parallel slices and due to Gruman and Amamou–Ben Farah in the case of concurrent slices. To cite this article: N. Ghiloufi, K. Dabbek, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Sur la transformation d'Abel–Radon de courants localement résiduels

The aim of this Note is to give a generalisation of the following theorem of Henkin and Passare: let Y be an analytic subvariety of pure codimension p in a linearly p-concave domain U, and ω a meromorphic q-form (q>0) on it; if the Abel–Radon transform $\text{A}\text{(ω∧[Y])}$, which is meromorphic on $\text{U}{}^1$, has a meromorphic prolongation to $\text{U}{}^1$, then Y extends to an analytic subvariety of $\text{U}$, and ω to a meromorphic form on it. The problem is to show the analogous statement when we replace ω∧[Y] by a current α of a more general type, called locally residual. We give the proof if α is of bidegree (N,1), or (q+1,1), 0<q<N in the particular case where $\text{A}\text{(α)=0}$. We conclude with some applications of the theorem. To cite this article: B. Fabre, C. R. Acad. Sci. Paris, Ser. I 338 (2004).