Lectures on Kähler Geometry

by Andrei Moroianu

Overview

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.

Product Details

ISBN-13:	9780521868914
Publisher:	Cambridge University Press
Publication date:	03/29/2007
Series:	London Mathematical Society Student Texts , #69
Pages:	182
Product dimensions:	5.98(w) x 9.02(h) x 0.55(d)

About the Author

Andrei Moroianu is a Researcher at CNRS and a Professor of Mathematics at Ecole Polytechnique.

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.

What People are Saying About This

From the Publisher

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

Editorial Reviews

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

From the Publisher