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Comptes Rendus Mathématique
Volume 357, n° 2
pages 130-142 (février 2019)
Doi : 10.1016/j.crma.2019.01.011
Received : 15 June 2018 ;  accepted : 4 February 2019
Courants résidus et opérateurs de Monge–Ampère
Residue currents and Monge–Ampère operators
 

Michel Méo
 Institut Élie Cartan de Lorraine, Université de Lorraine, boulevard des Aiguillettes, BP 70239, 54506 Vandœuvre-lès-Nancy, France 

Résumé

On étend au cas où le support est un sous-ensemble analytique éventuellement singulier le théorème de Federer de structure du courant résidu. Par ailleurs, on détermine la loi générale de transformation de la distribution   associée à une application holomorphe  . On arrive ainsi à l'interprétation cohomologique de la classe fondamentale associée à un cycle analytique effectif, qui n'est pas nécessairement localement intersection complète. Par cette même loi, on obtient une caractérisation des sous-ensembles algébriques de dimension pure de  , qui sont des intersections complètes. On caractérise aussi les intersections complètes de codimension q dans   en termes des solutions de l'équation de Monge–Ampère singulière dans  . Enfin, on exprime la condition sur la dimension des pôles de la fonction plurisousharmonique u impliquant que l'opérateur de Monge–Ampère   est d'ordre 0, pour tout courant positif fermé Q de bidimension  .

The full text of this article is available in PDF format.
Abstract

We extend to the case when the support is a possibly singular analytic subvariety the Federer theorem on the structure of the residue current. On another hand, we determine the general law of transformation of the distribution   associated with a holomorphic map  . In such a way, we arrive at the cohomological interpretation of the fundamental class of an effective analytic cycle, which is not necessarily a local complete intersection. By this same law, we obtain a characterization of the pure dimensional algebraic subsets of  , which are complete intersections. We also characterize the complete intersections of codimension q in   in terms of the solutions of the singular Monge–Ampère equation in  . Lastly, we express the condition on the dimension of the poles of the plurisubharmonic function u , so that the Monge–Ampère operator   has measure coefficients, for all closed positive current Q of bidimension  .

The full text of this article is available in PDF format.


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