Résumé
Un des buts de ce travail est d’illustrer de diverses manières l’efficacité des outils fondamentaux introduits par Pierre Lelong dans l’étude de l’Analyse Complexe et de la Géométrie analytique ou algébrique. Nous donnons d’abord une présentation détaillée du théorème d’extension L 2 de Ohsawa-Takegoshi, avec le même point de vue géométrique que celui introduit par L. Manivel. Ce faisant, nous simplifions la démarche de Ohsawa-Takegoshi et Manivel, et mettons en évidence une difficulté (non encore surmontée) dans l’argument invoqué par Manivel pour la régularité en bidegré (0, q), q ≥1. Nous donnons ensuite quelques applications frappantes du théorème d’extension, en particulier un théorème d’approximation des fonctions plurisousharmoniques par des logarithmes de fonctions holomorphes, préservant autant que possible les singularités et nombres de Lelong de la fonction plurisousharmonique donnée. L’étude des singularités de fonctions plurisousharmoniques se poursuit par un théorème de type Briançon-Skoda nouveau pour les faisceaux d’idéaux multiplicateurs de Nadel. En utilisant ce résultat et des idées de R Lazarsfeld, nous donnons finalement une preuve nouvelle d’un résultat récent de T. Fujita: la croissance du nombre des sections des multiples d’un fibré en droites gros sur une variété projective est donnée par la puissance d’intersection de plus haut degré de la partie numériquement effective dans la décomposition de Zariski du fibré.
Abstract
One of the goals of this work is to demonstrate in several different ways the strength of the fundamental tools introduced by Pierre Lelong for the study of Complex Analysis and Analytic or Algebraic Geometry. We first give a detailed presentation of the Ohsawa-Takegoshi L 2 extension theorem, inspired by a geometric viewpoint introduced by L. Manivel in 1993. Meanwhile, we simplify the original approach of the above authors, and point out a difficulty (yet to be overcome) in the regularity argument invoked by Manivel in bidegree (0, q), q ≥ 1. We then derive some striking consequences of the L2 extension theorem. In particular, we give an approximation theorem of plurisubharmonic functions by logarithms of holomorphic functions, preserving as much as possible the singularities and Lelong numbers of the given function. The study of plurisubharmonic singularities is pursued, leading to a new Briançon-Skoda type result concerning Nadel ‘s multiplier ideal sheaves. Using this result and some ideas of R Lazarsfeld, we finally give a new proof of a recent result of T. Fujita: the growth of the number of sections of multiples of a big line bundle is given by the highest power of the first Chern class of the numerically effective part in the line bundle Zariski decomposition.
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Demailly, JP. (2000). On the Ohsawa-Takegoshi-Manivel L 2 extension theorem. In: Dolbeault, P., Iordan, A., Henkin, G., Skoda, H., Trépreau, JM. (eds) Complex Analysis and Geometry. Progress in Mathematics, vol 188. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8436-5_3
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