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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.
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
32Q28 Stein manifolds
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