A viscosity approach to degenerate complex Monge-Ampère equations

Zeriahi, Ahmed

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 22 (2013) no. 4, pp. 843-913.

Ce qui suit reproduit les exposés de l’auteur à l’Ecole d’Hiver KAWA 3, qui s’est tenue à l’Université de Barcelone du 30 janvier au 3 février 2012. Le but principal était d’expliquer les techniques de viscosité et de les appliquer aux équations de Monge-Ampère complexes dégénérées.

Nous survolerons les techniques principales de l’approche par la viscosité, et montrerons comment les adapter aux équations de Monge-Ampère complexes dégénérées. Dans cette méthode, le point crucial est le « Principe de Comparaison » qui nous permet de prouver l’unicité des solutions sous des conditions de valeurs au bord.

Nous démontrerons un principe de comparaison de viscosité global pour les équations de Monge-Ampère complexes dégénérées sur les variétés compactes kählériennes et montrerons comment combiner les méthodes de viscosité et les méthodes de pluripotentiel pour obtenir des « versions continues » des Théorèmes de Calabi-Yau et Aubin-Yau dans certaines situations dégénérées. En particulier, nous démontrons l’existence de métriques de Kähler-Einstein singulières avec des potentiels continus sur les variétés de Kähler compactes normales avec des singularités modérées et un diviseur canonique ample ou trivial.

This is the content of the lectures given by the author at the winter school KAWA3 held at the University of Barcelona in 2012 from January 30 to February 3. The main goal was to give an account of viscosity techniques and to apply them to degenerate Complex Monge-Ampère equations.

We will survey the main techniques used in the viscosity approach and show how to adapt them to degenerate complex Monge-Ampère equations. The heart of the matter in this approach is the “Comparison Principle" which allows us to prove uniqueness of solutions with prescribed boundary conditions.

We will prove a global viscosity comparison principle for degenerate complex Monge-Ampère equations on compact Kähler manifolds and show how to combine Viscosity methods and Pluripotential methods to get “continuous versions" of the Calabi-Yau and Aubin-Yau Theorems in some degenerate situations. In particular we prove the existence of singular Kähler-Einstein metrics with continuous potentials on compact normal Kähler varieties with mild singularities and ample or trivial canonical divisor.

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