Lectures on Kähler manifolds. (English) Zbl 1101.53042

Ballmann, Werner

ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society Publishing House (ISBN 3-03719-025-6/pbk). x, 172 p. (2006).

This book is an introduction into the geometry of Kähler manifolds. Besides that, main aims of the author are cohomology of Kähler manifolds, formality of Kähler manifolds, Calabi Conjecture and some of its consequences, Gromov’s Kähler hyperbolicity and Kodaira embedding theorem.
Contents of this book; 1. Preliminaries (differential forms; Laplace operator; Hodge operator). 2. Complex manifolds. 3. Holomorphic vector bundles (Dolbeault Cohomology, Chern connection, line bundles). 4. Kähler manifolds. 5. Cohomology of Kähler manifolds (Lefschetz map and differentials, Lefschetz map and cohomology, the \(dd_c\)-lemma and formality, some vanishing theorems). 6. Ricci curvature and global structure (Ricci flat Kähler manifolds, nonnegative Ricci curvature, Ricci curvature and Laplace operator). 7. Calabi Conjecture. 8. Kähler hyperbolic spaces (Kähler hyperbolicity and Spectrum, non-vanishing of cohomology). 9. Kodaira embedding theorem and applications. Appendix A. Chern-Weil theory. Appendix B. Symmetric spaces (symmetric pairs, examples, Hermitian symmetric spaces). Appendix C. Remarks on differential operators (Dirac operators, \(L^2\)-de Rham cohomology, \(L^2\)-Dolbeault cohomology).
This book contains many examples and exercises as well as a large list of literature consisting of 103 references.

Reviewer: Nicolai K. Smolentsev (Kemerovo)

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
32Q15	Kähler manifolds
53-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)

Keywords:

Kähler manifolds; cohomology of Kähler manifolds; Calabi conjecture; Kodaira embedding theorem; Kähler hyperbolicity; symmetric spaces