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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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