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A theorem of Oka-Levi type for étale domains over projective manifolds. (Un théorème du type d’Oka-Levi pour les domaines étalés au dessus de variétés projectives.) (French) Zbl 0947.32010
Summary: We study spread domains $$\Pi:U\to V$$ over a projective manifold $$V$$ such that $$\Pi$$ is a Stein morphism, e.g., hull of meromorphy. We prove that such a domain is an existence domain of some holomorphic section $$s\in H^0(U,E^l)$$, where $$E=\Pi^*(H)$$, $$H$$ an ample line bundle on $$V$$. This is done by proving some line bundle convexity theorem for $$U$$. We deduce various results, e.g., a Lelong-Bremermann theorem for almost plurisubharmonic functions and a general Levi type theorem: Let $$U\to V$$ a locally pseudoconvex spread domain over a projective manifold, then $$U$$ is an almost domain of meromorphy, that is $$\widetilde U\setminus U=H$$ some hypersurface in $$\widetilde U$$, the hull of meromorphy of $$U$$. Hence, if $$W$$ is a general spread domain over $$V$$ then its pseudoconvex hull is obtained from its meromorphic hull minus some hypersurface.
##### MSC:
 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results 32Q28 Stein manifolds
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