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

Jean Pierre Demailly, János Kollár

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239 Scopus citations


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, three new examples of rigid Kähler-Einstein Del Pezzo surfaces with quotient singularities are obtained.

Original languageEnglish (US)
Pages (from-to)525-556
Number of pages32
JournalAnnales Scientifiques de l'Ecole Normale Superieure
Issue number4
StatePublished - Jul 2001

All Science Journal Classification (ASJC) codes

  • General Mathematics


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