02755nam 22004215a 450 991015763680332120161219234501.03-03719-667-X10.4171/167(CKB)3710000001001505(CH-001817-3)210-161219(PPN)197870120(EXLCZ)99371000000100150520161219j20170112 fy 0engurnn|mmmmamaatxtrdacontentcrdamediacrrdacarrierDegenerate Complex Monge-Ampère Equations[electronic resource] /Vincent Guedj, Ahmed ZeriahiZuerich, Switzerland European Mathematical Society Publishing House20171 online resource (496 pages)EMS Tracts in Mathematics (ETM)263-03719-167-8 Winner of the 2016 EMS Monograph Award! Complex Monge-Ampère equations have been one of the most powerful tools in Kähler geometry since Aubin and Yau's classical works, culminating in Yau's solution to the Calabi conjecture. A notable application is the construction of Kähler-Einstein metrics on some compact Kähler manifolds. In recent years degenerate complex Monge-Ampère equations have been intensively studied, requiring more advanced tools. The main goal of this book is to give a self-contained presentation of the recent developments of pluripotential theory on compact Kähler manifolds and its application to Kähler-Einstein metrics on mildly singular varieties. After reviewing basic properties of plurisubharmonic functions, Bedford-Taylor's local theory of complex Monge-Ampère measures is developed. In order to solve degenerate complex Monge-Ampère equations on compact Kähler manifolds, fine properties of quasi-plurisubharmonic functions are explored, classes of finite energies defined and various maximum principles established. After proving Yau's celebrated theorem as well as its recent generalizations, the results are then used to solve the (singular) Calabi conjecture and to construct (singular) Kähler-Einstein metrics on some varieties with mild singularities. The book is accessible to advanced students and researchers of complex analysis and differential geometry.Complex analysisbicsscSeveral complex variables and analytic spacesmscComplex analysisSeveral complex variables and analytic spaces515.932-xxmscGuedj Vincent524796Zeriahi Ahmedch0018173BOOK9910157636803321Degenerate Complex Monge-Ampère Equations2564449UNINA