Complex Monge-Ampère equations and geodesics in the space of Kähler metrics. Lecture notes (Q639220)

This set of lecture notes is devoted to explore various aspects of the homogeneous complex Monge-Ampère equations, in particular (1) regularity properties of the solution of such equations and (2) their links with the geodesics in the space of Kähler metrics (on a compact Käahler manifold). The book covers a wide range of questions in the topic. Some of them are classical such as the study of the Monge-Ampère operator in strictly pseudoconvex domains of \(\mathbb{C}^n\) and the solution of the Calabi-Yau conjecture. Others consist in recent advances: existence of geodesics in the space of Kähler metrics and their approximations by their finite dimensional analogues. This book consists in a very good introduction to the study of Monge-Ampère equations in various contexts and its reading should be adviced to anyone who would learn recent advances in the subject. The main chapters, which are reviewed individually, are as follows: ``Dirichlet problem in domains of \(\mathbb{C}^n\)'' by the editor and \textit{A. Zeriahi} [Zbl 1231.32029]; ``Geometric properties of maximal psh functions'' by \textit{R. Dujardin} and the editor [Zbl 1231.32027]; ``Probabilistic approach to regularity'' by \textit{F. Delarue} [Zbl 1231.32026]; ``The Calabi-Yau theorem'' by \textit{Z. Błocki} [Zbl 1231.32017]; ``The Riemannian space of Kähler metrics'' by \textit{B. Kolev} [Zbl 1231.32016]; ``Monge-Ampère equations on complex manifolds with boundary'' by \textit{S. Boucksom} [Zbl 1231.32025]; ``Bergman geodesics'' by \textit{R. Berman} and \textit{J. Keller} [Zbl 1231.32002].