Nombres de Lelong généralisés, théorèmes d’intégralité et d’analyticité. (Generalized Lelong numbers, integrability and analyticity theorems). (French) Zbl 0629.32011

Let X be a complex Stein space, T a closed positive current of bidimension (p,p) on X and \(\phi:X\to [-\infty,+\infty [\) an exhaustive plurisubharmonic function. The author’s generalized Lelong number \(\nu\) (T,\(\phi)\) is defined as the mass of the measure \(T\wedge (dd^ c\phi)^ p\) carried by the polar set \(\phi^{-1}(-\infty)\) and is obtained by means of the Monge-Ampère operator of E. Bedford and B. A. Taylor [ibid. 149, 1-40 (1982; Zbl 0547.32012)]. \(\nu\) (T,\(\phi)\) generalizes the classical P. Lelong [“Plurisubharmonic functions and positive differential forms” (1969; Zbl 0195.116)] and C. O. Kiselman’s numbers. The author establishes that \(\nu\) (T,\(\phi)\) depends only on the behaviour of \(\phi\) in a neighbourhood of the poles. The use of \(\nu\) (T,\(\phi)\) allows him to obtain very simple proofs of classical results on Lelong numbers, e.g. that these numbers are invariant with respect to local coordinate transformations [cf. Y. T. Siu, Invent. Math. 27, 53-156 (1974; Zbl 0289.32003)] and also on P. Thie’s [Math. Ann. 172, 269-312 (1967; Zbl 0158.328)] theorem showing that the Lelong number of an analytic set X coincides to the algebraic multiplicity of Y at x. Finally, the author obtains a generalization of Siu’s theorem on the analyticity of the level sets associated to Lelong numbers, his result containing as a particular case a recent theorem of C. O. Kiselman on directional Lelong numbers.
Reviewer: P.Caraman


32E10 Stein spaces
32U05 Plurisubharmonic functions and generalizations
32C30 Integration on analytic sets and spaces, currents
31C10 Pluriharmonic and plurisubharmonic functions
31C15 Potentials and capacities on other spaces
Full Text: DOI


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