Semi-continuity of complex singularity exponents and Kahler-Einstein metrics on Fano orbifolds

We introduce complex singularity exponents of plurisubharmonic functions an d 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 algebra ic case, but it also takes into account more quantitative informations of g reat interest for complex analysis and complex differential geometry. We gi ve as an application a new derivation of criteria for the existence of Kahl er-Einstein metrics on certain Fano orbifolds, following Nadel's original i deas (but with a drastic simplication in the technique, once the semi-conti nuity result is taken for granted). In this way, three new examples of rigi d Kahler-Einstein Del Pezzo surfaces with quotient singularities are obtain ed. (C) 2001 Editions scientifiques et medicales Elsevier SAS.