Guedj V., Zeriahi A. Degenerate Complex Monge-Ampère Equations

European Mathematical Society, 2017. — 498 p. — (EMS Tracts in Mathematics 26). — ISBN-13 9783037191675.Вырожденные комплексные уравнения Monge–AmpèreComplex 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 local theory
Plurisubharmonic functions
Positive currents
The complex Monge–Amp`ere operator
The Monge–Amp`ere capacity
The Dirichlet problem
Viscosity solutions
Pluripotential theory on compact manifolds
CompactK¨ahler manifolds
Quasi-plurisubharmonic functions
Envelopes and capacities
Finite energy classes
Solving complex Monge–Amp`ere equations
The variational approach
Uniform a priori estimates
The viscosity approach
Smooth solutions
Singular K¨ahler–Einstein metrics
Canonical metrics
Singularities and the Minimal Model Program