Kahler geometry is a beautiful and intriguing area of mathematics, of substantial research interest to both mathematicians and physicists. This self-contained graduate text provides a concise and accessible introduction to the topic. The book begins with a review of basic differential geometry, before moving on to a description of complex manifolds and holomorphic vector bundles. Kahler manifolds are discussed from the point of view of Riemannian geometry, and Hodge and Dolbeault theories are outlined, together with a simple proof of the famous Kahler identities. The final part of the text studies several aspects of compact Kahler manifolds: the Calabi conjecture, Weitzenboeck techniques, Calabi-Yau manifolds, and divisors. All sections of the book end with a series of exercises and students and researchers working in the fields of algebraic and differential geometry and theoretical physics will find that the book provides them with a sound understanding of this theory.
Andrei Moroianu is a Researcher at CNRS and a Professor of Mathematics at Ecole Polytechnique.
Introduction; Part I. Basics on Differential Geometry: 1. Smooth manifolds; 2. Tensor fields on smooth manifolds; 3. The exterior derivative; 4. Principal and vector bundles; 5. Connections; 6. Riemannian manifolds; Part II. Complex and Hermitian Geometry: 7. Complex structures and holomorphic maps; 8. Holomorphic forms and vector fields; 9. Complex and holomorphic vector bundles; 10. Hermitian bundles; 11. Hermitian and Kahler metrics; 12. The curvature tensor of Kahler manifolds; 13. Examples of Kahler metrics; 14. Natural operators on Riemannian and Kahler manifolds; 15. Hodge and Dolbeault theory; Part III. Topics on Compact Kahler Manifolds: 16. Chern classes; 17. The Ricci form of Kahler manifolds; 18. The Calabi-Yau theorem; 19. Kahler-Einstein metrics; 20. Weitzenboeck techniques; 21. The Hirzebruch-Riemann-Roch formula; 22. Further vanishing results; 23. Ricci-flat Kahler metrics; 24. Explicit examples of Calabi-Yau manifolds; Bibliography; Index.