A viscosity approach to degenerate complex Monge-Ampère equations.
Résumé
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 Febrary 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 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 Calabi-Yau and Aubin-Yau Theorems in some degenerate situations. In particular we show existence of singular Kähler-Einstein metrics with continuous potentials on compact normal Kähler varieties with mild singularities and ample or trivial canonical divisor.