In this note a necessary and sufficient condition for a compact complex space X to be Moishezon is obtained; it can be seen as the existence of a line bundle L on X such that, for some point x ∈ X, the first cohomology groups of X with values respectively in L ⊗ mx and L ⊗ mx2, vanish. (Here mx denotes the ideal sheaf at x).
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...