# Lectures on Kähler Geometry (London Mathematical Society Student Texts)

Cambridge University Press, 3/29/2007

EAN 9780521688970, ISBN10: 0521688973

Paperback, 182 pages, 24.4 x 17 x 1 cm

Language: English

KÃƒÂ¤hler 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. KÃƒÂ¤hler 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 KÃƒÂ¤hler identities. The final part of the text studies several aspects of compact KÃƒÂ¤hler manifolds: the Calabi conjecture, WeitzenbÃƒÂ¶ck techniques, CalabiÃ¢â‚¬â€œYau 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 KÃƒÂ¤hler metrics

12. The curvature tensor of KÃƒÂ¤hler manifolds

13. Examples of KÃƒÂ¤hler metrics

14. Natural operators on Riemannian and KÃƒÂ¤hler manifolds

15. Hodge and Dolbeault theory

Part III. Topics on Compact KÃƒÂ¤hler Manifolds

16. Chern classes

17. The Ricci form of KÃƒÂ¤hler manifolds

18. The CalabiÃ¢â‚¬â€œYau theorem

19. KÃƒÂ¤hlerÃ¢â‚¬â€œEinstein metrics

20. WeitzenbÃƒÂ¶ck techniques

21. The HirzebruchÃ¢â‚¬â€œRiemannÃ¢â‚¬â€œRoch formula

22. Further vanishing results

23. RicciÃ¢â‚¬â€œflat KÃƒÂ¤hler metrics

24. Explicit examples of CalabiÃ¢â‚¬â€œYau manifolds

Bibliography

Index.