The theory of complex manifolds overlaps with several branches of mathematics, including differential geometry, algebraic geometry, several complex variables, global analysis, topology, algebraic number theory, and mathematical physics. Complex manifolds provide a rich class of geometric objects, for example the (common) zero locus of any generic set of complex polynomials is always a complex manifold. Yet complex manifolds behave differently than generic smooth manifolds; they are more coherent and fragile. The rich yet restrictive character of complex manifolds makes them a special and interesting object of study.

This book is a self-contained graduate textbook that discusses the differential geometric aspects of complex manifolds. The first part contains standard materials from general topology, differentiable manifolds, and basic Riemannian geometry. The second part discusses complex manifolds and analytic varieties, sheaves and holomorphic vector bundles, and gives a brief account of the surface classification theory, providing readers with some concrete examples of complex manifolds. The last part is the main purpose of the book; in it, the author discusses metrics, connections, curvature, and the various roles they play in the study of complex manifolds. A significant amount of exercises are provided to enhance student comprehension and practical experience.

Titles in this series are co-published with International Press, Cambridge, MA.

Readership

Graduate students and research mathematicians interested in differential geometry.

Reviews

"Considering the vast amount of material covered and part of the material once used in summer school ... the presentation is precise and lucid ... If one has some background or previous exposure to some of the material in the book, studying this book would be really enjoyable and one could learn a lot from it. It is also a very good reference book."

-- Mathematical Reviews

Table of Contents

• Part 1 introduction
• Differentiable manifolds and vector bundles
• Metric, connection, and curvature
• The geometry of complete Riemannian manifolds

• Part 2 introduction
• Complex manifolds and analytic varieties
• Holomorphic vector bundles, sheaves and cohomology
• Compact complex surfaces

• Part 3 introduction
• Hermitian and Kähler metrics
• Compact Kähler manifolds
• Kähler geometry
• Bibliography
• Index