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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

MSC:
 32E10 Stein spaces, Stein manifolds 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
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References:
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