Lectures on Kähler Geometry

This graduate text provides a concise and self-contained introduction to Kähler geometry.

Andrei Moroianu (Author)

9780521868914, Cambridge University Press

Hardback, published 29 March 2007

182 pages, 131 exercises
22.9 x 15.2 x 1.4 cm, 0.44 kg

"A concise and well-written modern introduction to the subject."
Tatyana E. Foth, Mathematical Reviews

Kähler 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. Kähler 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 Kähler identities. The final part of the text studies several aspects of compact Kähler manifolds: the Calabi conjecture, Weitzenböck 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.

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 Kähler metrics
12. The curvature tensor of Kähler manifolds
13. Examples of Kähler metrics
14. Natural operators on Riemannian and Kähler manifolds
15. Hodge and Dolbeault theory
Part III. Topics on Compact Kähler Manifolds: 16. Chern classes
17. The Ricci form of Kähler manifolds
18. The Calabi–Yau theorem
19. Kähler–Einstein metrics
20. Weitzenböck techniques
21. The Hirzebruch–Riemann–Roch formula
22. Further vanishing results
23. Ricci–flat Kähler metrics
24. Explicit examples of Calabi–Yau manifolds
Bibliography
Index.

Subject Areas: Mathematical physics [PHU], Geometry [PBM]