Pluripotential theory on compact Hermitian manifolds

— In this survey article we collect the basic results in pluripotential theory in the setting of compact Hermitian manifolds. In particular we discuss in detail the corresponding capacity theory, various comparison principles, and the solution of the Hermitian counterpart of the CalabiYau equation. Introduction Pluripotential theory in the setting of compact Kähler manifolds has proven to be a very effective tool in the study of degeneration of metrics in geometrically motivated problems (see [39, 40, 18, 42], which is by far incomplete list of the literature on the subject). Usually in such a setting singular Kähler metrics do appear as limits of smooth ones. Then pluripotential theory provides a natural background for defining the singular volume forms associated to such metrics. More importantly it provides useful information on the behavior of the Kähler potentials exactly along the singularity locus, which is hard to achieve by standard PDE techniques. On the other hand (∗) Reçu le 03/12/2014, accepté le 16/06/2015 (1) Department of Mathematics and Computer Science, Jagiellonian University, 30-409 Kraków, ul. Lojasiewicza 6, Poland slawomir.dinew@im.uj.edu.pl Article proposé par Vincent Guedj.