The existence problem of <i>S</i>-plurisubharmonic currents

Ahmad K. Al Abdulaali; Hassine El Mir

doi:10.1016/j.crma.2015.04.011

Complex analysis

The existence problem of S-plurisubharmonic currents

Presented by: Jean-Pierre Demailly

Ahmad K. Al Abdulaali ¹; Hassine El Mir ¹

¹ The Department of Mathematics, College of Science, King Faisal University, P.O. Box 380, Post Code 31982, Al-Ahsaa, Saudi Arabia

Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 605-610.

Abstract (Original language)
Résumé

In this paper, we prove the existence of the trivial extension of S-plurisubharmonic currents of bidimension $(p, p)$ defined outside an obstacle A of Hausdorff measure $H_{2 p} (A) = 0$ . Furthermore, a valid definition of the current $d g \land d^{c} g \land T$ is achieved for every positive closed current T and plurisubharmonic function g. The above results rely on an improvement of a classical result due to Demailly on the Monge–Ampère operator with a sharp condition on the Hausdorff measure.

Dans cette Note, nous montrons l'existence de l'extension triviale des courants S-plurisousharmoniques de bi-dimension $(p, p)$ , définis en-dehors d'un obstacle A de mesure de Hausdorff $H_{2 p} (A) = 0$ . De plus, nous montrons que le courant $d g \land d^{c} g \land T$ est bien défini, pour tout courant positif fermé T et toute fonction plurisousharmonique g. Ces résultats reposent sur un relâchement de la condition de nullité d'une mesure de Hausdorff, dans un résultat classique de Demailly sur l'opérateur de Monge–Ampère.

Received: 2015-02-23
Accepted: 2015-04-21
Published online: 2015-04-30

DOI: 10.1016/j.crma.2015.04.011

Author's affiliations:

Ahmad K. Al Abdulaali ¹; Hassine El Mir ¹

¹ The Department of Mathematics, College of Science, King Faisal University, P.O. Box 380, Post Code 31982, Al-Ahsaa, Saudi Arabia

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     author = {Ahmad K. Al Abdulaali and Hassine El Mir},
     title = {The existence problem of {\protect\emph{S}-plurisubharmonic} currents},
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     language = {en},
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Ahmad K. Al Abdulaali; Hassine El Mir. The existence problem of S-plurisubharmonic currents. Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 605-610. doi : 10.1016/j.crma.2015.04.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.04.011/

References
Cited by

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[10] B. Shiffman On the removal of singularities of analytic sets, Mich. Math. J., Volume 15 (1968), pp. 111-120

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