AMS/IP Studies in Advanced Mathematics 2000; 264 pp; softcover Volume: 18 Reprint/Revision History: reprinted with corrections 2002 ISBN10: 0821829602 ISBN13: 9780821829608 List Price: US$60 Member Price: US$48 Order Code: AMSIP/18.S
 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 selfcontained 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 copublished with International Press of Boston, Inc., 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 Reimannian geometry  Part 1 introduction
 Differentiable manifolds and vector bundles
 Metric, connection, and curvature
 The geometry of complete Riemannian manifolds
Complex manifolds  Part 2 introduction
 Complex manifolds and analytic varieties
 Holomorphic vector bundles, sheaves and cohomology
 Compact complex surfaces
Kähler geometry  Part 3 introduction
 Hermitian and Kähler metrics
 Compact Kähler manifolds
 Kähler geometry
 Bibliography
 Index
