Degenerate complex Monge--Ampère equations
By: Vincent Guedj, Ahmed Zeriahi
Contributors: Zeriahi, Ahmed
Series: EMS tracts in mathematics -- 26.; EMS tracts in mathematics -- 26.
Publisher: European Mathematical Society Publishing House
Published: 2017
Place of publication: Zürich, Switzerland
Language: eng
Number of pages: 472
ISBN-13: 9783037191675
Subjects: Monge-Ampère equations; Pluripotential theory; Plurisubharmonic functions; MATHEMATICS / Differential Equations / Partial
Description: 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.
Notes: Includes bibliographical references (pages 451-467) and index.