Degenerate omplex Monge Ampère equations

Degenerate omplex Monge-Ampère equations over ompa t Kähler manifolds Jean-Pierre Demailly and Nefton Pali Abstra t We prove the existen e and uniqueness of the solutions of some very general type of degenerate omplex Monge-Ampère equations, and investigate their regularity. This type of equations are pre isely what is needed in order to onstru t Kähler-Einstein metri s over ir- redu ible singular Kähler spa es with ample or trivial anoni al sheaf and singular Kähler-Einstein metri s over varieties of general type. Contents 1 Introdu tion 1 2 General L∞-estimates for the solutions 5 3 Currents with Bedford-Taylor type singularities 23 4 Uniqueness of the solutions 35 5 Generalized Kodaira lemma 43 6 Existen e and higher order regularity of solutions 46 7 Appendix 60 1 Introdu tion In a elebrated paper [Yau? published in 1978, Yau solved the Calabi onje - ture. As is well known, the problem of pres ribing the Ri i urvature an be formulated in terms of non-degenerate omplex Monge-Ampère equations. Key words : Complex Monge-Ampère equations, Kähler-Einstein metri s, Closed posi- tive urrents, Plurisubharmoni fun tions, Capa ities, Orli z spa es. AMS Classi ation : 53C25, 53C55, 32J15.

• degenerate omplex


• ontinuous lo


• monge-ampère equations


• al potentials


• over ompa


• kähler-einstein metri


• general l∞