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On the Ohsawa-Takegoshi-Manivel $$L^2$$ extension theorem. (English) Zbl 0959.32019
Dolbeault, P. (ed.) et al., Complex analysis and geometry. Proceedings of the international conference in honor of Pierre Lelong on the occasion of his 85th birthday, Paris, France, September 22-26, 1997. Basel: Birkhäuser. Prog. Math. 188, 47-82 (2000).
The Ohsawa-Takegoshi-Manivel $$L^2$$ extension theorem addresses the following basic problem: Let $$Y$$ be a complex analytic submanifold of a complex manifold $$X$$; given a holomorphic function $$f$$ on $$Y$$ satisfying suitable $$L^2$$ conditions on $$Y,$$ find a holomorphic extension $$F$$ of $$f$$ to $$X,$$ together with a good $$L^2$$ estimate for $$F$$ on $$X.$$
The first satisfactory solution of this problem has been obtained by T. Ohsawa and K. Takegoshi. The author follows here a more geometric approach due to L. Manivel, which provides a more general extension theorem in the framework of vector bundles and higher cohomology groups. The first goal of this note is to simplify further Manivel’s approach, as well as to point out a technical difficulty in Manivel’s proof. The author uses a simplified and slightly extended version of the original Ohsawa-Takegoshi a priori inequality. Then the Ohsawa-Takegoshi-Manivel extension theorem is applied to solve several important problems of complex analysis or geometry. The first of these is an approximation theorem for plurisubharmonic functions. It is shown that the approximation can be made with a uniform convergence of the Lelong numbers of the holomorphic functions towards those of the given plurisubharmonic function. This result contains as a special case Siu’s theorem on the analyticity of Lelong number sublevel sets. By combining some of the results provided by the proof of that approximation theorem with Skoda’s $$L^2$$ estimates for the division of holomorphic functions, a Briançon-Skoda type theorem for Nadel’s multiplier ideal sheaves is obtained. Using this result and some ideas of R. Lazarsfeld, it is obtained 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.
For the entire collection see [Zbl 0940.00031].

##### MSC:
 32D15 Continuation of analytic objects in several complex variables 32U05 Plurisubharmonic functions and generalizations