This book offers a systematic description of the basic properties of Kähler manifolds together with a few more advanced topics chosen from differential geometry and global analysis. The first three chapters review basic facts about the Laplace and the Hodge operators on differential forms, vector fields and forms on complex manifolds, and the Dolbeault cohomology and connections on holomorphic vector bundles. Kähler manifolds and their cohomology are studied in the next two chapters (including the corresponding Levi-Civita connection, its curvature, the Ricci tensor, Killing fields, the Lefschetz theorem, the Hodge-Riemann bilinear relations, the Hodge index theorem and the Kodaira vanishing theorem). The next four chapters contain some advanced topics: a few relations between the behaviour of the Ricci tensor and global properties of the manifold, the Calabi conjecture (together with substantial parts of the proof), the Gromov results on Kähler hyperbolic spaces and the Kodaira embedding theorem.
The book is written in a systematic, precise and understandable style. The background needed includes parts of differential geometry (vector bundles and connections, curvature and holonomy) and global analysis (Sobolev embedding theorems, regularity and the Hodge theory for elliptic partial differential equations). The Chern-Weil theory of characteristic classes and basic facts on symmetric spaces are comfortably summarized in appendices. The book will be very useful both for mathematicians and theoretical physicists who need Kähler manifolds in their research.