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

Abstract

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

Cauchy–Riemann meet Monge–Ampère

“…We will start with the following recent result of Berndtsson [10] (proved by Favre and Jonsson [54] in dimension 2) confirming the openness conjecture of Demailly-Kollár [46].…”

“…It has found many applications in complex and algebraic geometry but it can be also very useful to study singularities of plurisubharmonic functions. For example, it turns out that two main results in this area, the theorem of Siu [107] on analyticity of level sets of Lelong numbers and the openness conjecture of Demailly and Kollár [46] follow relatively easily from the Ohsawa-Takegoshi theorem. The simple proof of the Siu theorem was found by Demailly [45] who devised a special approximation of an arbitrary plurisubharmonic function by smooth ones with possibly analytic singularities.…”

Strong openness of multiplier ideal sheaves and optimal L 2 extension

“…In [23,25], we prove two conjectures posed by Demailly-Kollár (see [15]) and Jonsson-Mustatȃ (see [28]) respectively by using the following result. have positive lower bounds independent of r ∈ (0, 1).…”

“…When I(ϕ) is trivial, Demailly's strong openness conjecture degenerates to the openness conjecture posed in [15] which was proved by Berndtsson [5] (dimension two case was proved by Favre and Jonsson [18]). …”

Canonical and log canonical thresholds of Fano complete intersections

“…The importance of canonical and log canonical thresholds is connected with their applications to the complex differential geometry and birational geometry. Tian, Nadel, Demailly and Kollár showed in [7,12,20], that the inequality…”