TY - JOUR
T1 - Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds
AU - Demailly, Jean Pierre
AU - Kollár, János
PY - 2001/7
Y1 - 2001/7
N2 - 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.
AB - 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.
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U2 - 10.1016/S0012-9593(01)01069-2
DO - 10.1016/S0012-9593(01)01069-2
M3 - Article
AN - SCOPUS:0000817732
SN - 0012-9593
VL - 34
SP - 525
EP - 556
JO - Annales Scientifiques de l'Ecole Normale Superieure
JF - Annales Scientifiques de l'Ecole Normale Superieure
IS - 4
ER -