Complex analysis
Logarithmic potentials on Pn
Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 283-287.

We study the projective logarithmic potential Gμ of a probability measure μ on the complex projective space Pn. We prove that the range of the operator μGμ is contained in the (local) domain of definition of the complex Monge–Ampère operator acting on the class of quasi-plurisubharmonic functions on Pn with respect to the Fubini–Study metric. Moreover, when the measure μ has no atom, we show that the complex Monge–Ampère measure of its logarithmic potential is an absolutely continuous measure with respect to the Fubini–Study volume form on Pn.

On étudie le potentiel logarithmique projectif Gμ d'une mesure de probabilité μ sur l'espace projectif complexe Pn. On établit que l'image de l'opérateur μGμ est contenue dans le domaine de définition (local) de l'opérateur de Monge–Ampère complexe agissant sur les fonctions quasi-plurisousharmoniques dans Pn par rapport à la métrique de Fubini–Study. Si μ n'a pas d'atomes, on montre que la mesure de Monge–Ampère complexe du potentiel logarithmique de μ est absolument continue par rapport à la forme volume de Fubini–Study de Pn.

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DOI: 10.1016/j.crma.2018.02.004
Assila, Fatima Zahra 1

1 Université Ibn Tofail, avenue de l'Université, Kénitra, Maroc
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Assila, Fatima Zahra. Logarithmic potentials on $ {\mathbb{P}}^{n}$. Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 283-287. doi : 10.1016/j.crma.2018.02.004. http://www.numdam.org/articles/10.1016/j.crma.2018.02.004/

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