We introduce complex singularity exponents of plurisubharmonic functions and prove a general semi-continuity result for them. This concept contains as a special case several similar concepts which have been considered e.g. by Arnold and Varchenko, mostly for the study of hypersurface singularities. The plurisubharmonic version is somehow based on a reduction to the algebraic case, but it also takes into account more quantitative informations of great interest for complex analysis and complex differential geometry. We give as an application a new derivation of criteria for the existence of Kähler-Einstein metrics on certain Fano orbifolds, following Nadel's original ideas (but with a drastic simplication in the technique, once the semi-continuity result is taken for granted). In this way, 3 new examples of rigid Kähler-Einstein Del Pezzo surfaces with quotient singularities are obtained.Résumé. Nous introduisons les exposants de singularités complexes des fonctions plurisousharmoniques et démontrons un théorème de semi-continuité général pour ceux-ci. Le conceptétudié contient comme cas particulier des concepts voisins qui ontété considérés par exemple par Arnold et Varchenko, principalement pour l'étude des singularités d'hypersurfaces. La version plurisousharmonique repose en définitive sur une réduction au cas algébrique, mais elle prend aussi en compte des informations quantitatives d'un grand intérêt pour l'analyse complexe et la géométrie différentielle complexe. Nous décrivons en application une nouvelle approche des critères d'existence de métriques Kähler-Einstein pour les variétés de Fano, en nous inspirant des idées originales de Nadel -mais avec des simplifications importantes de la technique, une fois que le résultat de semi-continuité est utilisé comme outil de base. Grâceà ces critères, nous obtenons trois nouveaux exemples de surfaces de Del Pezzoà singularités quotients, rigides, possédant une métrique de Kähler-Einstein. 2 J.-P. Demailly, J. Kollár, Semi-continuity of complex singularity exponents Contents
Abstract. The main purpose of this paper is to generalize the celebrated L 2 extension theorem of Ohsawa-Takegoshi in several directions : the holomorphic sections to extend are taken in a possibly singular hermitian line bundle, the subvariety from which the extension is performed may be non reduced, the ambient manifold is Kähler and holomorphically convex, but not necessarily compact.
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