Lectures on Kähler Geometry
Lectures on Kähler Geometry
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.
- The first graduate-level text on Kähler geometry, providing a concise introduction for both mathematicians and physicists with a basic knowledge of calculus in several variables and linear algebra
- Over 130 exercises and worked examples
- Self-contained and presents varying viewpoints including Riemannian, complex and algebraic
Product details
March 2007Paperback
9780521688970
182 pages
229 × 153 × 12 mm
0.266kg
131 exercises
Available
Table of Contents
- 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.
