The complex Monge-Ampère equation and pluripotential theory / Sławomir Kołodziej

A collection of results on the existence and stability of weak solutions of complex Monge-Ampére equation proved by applying pluripotential theory methods and obtained in past three decades. Firstly introducing basic concepts and theorems of pluripotential theory, then the Dirichlet problem for the complex Monge-Ampére equation is studied. The main goal is to give possibly detailed description of the nonnegative Borel measures which on the right hand side of the equation give rise to plurisubharmonic solutions satisfying additional requirements such as continuity, boundedness or some weaker ones. In the last part the methods of pluripotential theory are implemented to prove the existence and stability of weak solutions of the complex Monge-Ampére equation on compact Kähler manifolds. This is a generalization of the Calabi-Yau theorem