Complex Monge-Ampère equattions and Kähler geometry. – MACK

Several fundamental questions of Kähler geometry boil down to solving certain complex Monge-Ampère equations. This is for instance
the case of the Kähler- Einstein equation solved by Aubin and Yau in the 70's when the curvature is non-positive.

The case of positive curvature has motivated many works since then (Siu, Yau, Tian, Nadel, Donaldson, Mabuchi, Phong-Sturm, etc) and
it is still widely open in higher dimensions. This is also the case for the study of the Kähler-Ricci flow.

It has become clear in the last few years that it is important to study these objects on singular varieties. Desingularizing the variety tranforms
the corresponding complex Monge-Ampère equations into degenerate ones: the metrics under consideration are now only semi-positive, which
leads to more singular solutions than in the classical situation. It is therefore necessary to develop tools which allow to treat such
degenerate situations.

Meanwhile the local pluripotential theory has gained considerable interest, after Bedford and Taylor laid down the bases of the theory.
They succeeded in defining and understanding the complex Monge-Ampère operator acting on any bounded pluri-subharmonic function.

Since then several authors have pushed the analysis further (Demailly, Fornaess-Sibony, Cegrell, Kolodziej,etc) so as to allow quite singular
functions within the domain of definition of this non-linear operator. One of the great successes of the theory lies in the full characterization of
the range of the complex Monge-Ampère operator on the classes of singular pluri-subharmonic functions of finite energy (Cegrell, Guedj-Zeriahi).

The tools developed have further allowed Kolodziej to give an alternative proof of Yau's celebrated uniform a priori estimate, opening the road
to further generalizations. The latter have been considered only recently, due to the complexity and scale of the preliminary material needed to go further.

Our aim is to constitute a small group of French experts with complementary backgrounds in order to start a sytematic study of these
degenerate complex Monge-Ampère equations on compact Kähler manifolds and apply them to complex geometry.