| Résume | On (X, ω) compact Kähler manifold, I will show that the Monge-Ampère operator is an homeomorphism between a set XA of certain ω-plurisubharmonic functions whose singularities are encoded in a total ordered family A ⊂ PSH(X, ω) representing some singularity types and a set YA of measures. The elements of these sets are characterized by having relative finite energy, so they have natural strong topologies which make the energies continuous. Then I will expose how to deal with complex Monge-Ampère equations with prescribed singularities through a continuity method which also involves the singularities types. As application and main motivation, I will also present how in the Fano case the existence of Kähler-Einstein metrics with prescribed singularities is related to the existence of the genuine Kähler-Einstein metrics. |
