Structures On Complex Manifolds

This book is an introduction to the theory of complex manifolds and their deformations. Deformation of the complex structure of Riemann surfaces is an idea which goes back to Riemann who, in his famous memoir on Abelian functions published in 1857, calculated the number of effective parameters on which the deformation depends. Since the publication of Riemann's memoir, questions concerning the deformation of the complex structure of Riemann surfaces have never lost their interest. The deformation of algebraic surfaces seems to have been considered first by Max Noether in 1888 (M. Noether: Anzahl der Modulen einer Classe algebraischer Fliichen, Sitz. K6niglich. Preuss. Akad. der Wiss. zu Berlin, erster Halbband, 1888, pp. 123-127). However, the deformation of higher dimensional complex manifolds had been curiously neglected for 100 years. In 1957, exactly 100 years after Riemann's memoir, Frolicher and Nijenhuis published a paper in which they studied deformation of higher dimensional complex manifolds by a differential geometric method and obtained an important result. (A. Fr61icher and A. Nijenhuis: A theorem on stability of complex structures, Proc. Nat. Acad. Sci., U.S.A., 43 (1957), 239-241).

A brand new appendix by Oscar Garcia-Prada graces this third edition of a classic work. In developing the tools necessary for the study of complex manifolds, this comprehensive, well-organized treatment presents in its opening chapters a detailed survey of recent progress in four areas: geometry (manifolds with vector bundles), algebraic topology, differential geometry, and partial differential equations. Wells’s superb analysis also gives details of the Hodge-Riemann bilinear relations on Kahler manifolds, Griffiths's period mapping, quadratic transformations, and Kodaira's vanishing and embedding theorems. Oscar Garcia-Prada’s appendix gives an overview of the developments in the field during the decades since the book appeared.

Serves as an introduction to the Kodaira-Spencer theory of deformations of complex structures. Based on lectures given by Kunihiko Kodaira at Stanford University in 1965-1966, this book gives the original proof of the Kodaira embedding theorem, showing that the restricted class of Kahler manifolds called Hodge manifolds is algebraic.

From the reviews of the second edition: "The new methods of complex manifold theory are very useful tools for investigations in algebraic geometry, complex function theory, differential operators and so on. The differential geometrical methods of this theory were developed essentially under the influence of Professor S.-S. Chern's works. The present book is a second edition... It can serve as an introduction to, and a survey of, this theory and is based on the author's lectures held at the University of California and at a summer seminar of the Canadian Mathematical Congress.... The text is illustrated by many examples... The book is warmly recommended to everyone interested in complex differential geometry." #Acta Scientiarum Mathematicarum, 41, 3-4#

In developing the tools necessary for the study of complex manifolds, this comprehensive, well-organized treatment presents in its opening chapters a detailed survey of recent progress in four areas: geometry (manifolds with vector bundles), algebraic topology, differential geometry, and partial differential equations. Subsequent chapters then develop such topics as Hermitian exterior algebra and the Hodge *-operator, harmonic theory on compact manifolds, differential operators on a Kahler manifold, the Hodge decomposition theorem on compact Kahler manifolds, the Hodge-Riemann bilinear relations on Kahler manifolds, Griffiths's period mapping, quadratic transformations, and Kodaira's vanishing and embedding theorems. The third edition of this standard reference contains a new appendix by Oscar Garcia-Prada which gives an overview of certain developments in the field during the decades since the book first appeared. From reviews of the 2nd Edition: "..the new edition of Professor Wells' book is timely and welcome...an excellent introduction for any mathematician who suspects that complex manifold techniques may be relevant to his work." - Nigel Hitchin, Bulletin of the London Mathematical Society "Its purpose is to present the basics of analysis and geometry on compact complex manifolds, and is already one of the standard sources for this material." - Daniel M. Burns, Jr., Mathematical Reviews

Easily accessible Includes recent developments Assumes very little knowledge of differentiable manifolds and functional analysis Particular emphasis on topics related to mirror symmetry (SUSY, Kaehler-Einstein metrics, Tian-Todorov lemma)

Contents: Riemannian ManifoldsSubmanifolds of Riemannian ManifoldsComplex ManifoldsSubmanifolds of Kaehlerian ManifoldsContact ManifoldsSubmanifolds of Sasakian Manifoldsf-StructuresProduct ManifoldsSubmersions Readership: Mathematicians. Keywords:Riemannian Manifold;Submanifold;Complex Manifold;Contact Manifold;Kaehlerian Manifold;Sasakian Manifold;Anti-Invariant Submanifold;CR Submanifold;Contact CR Submanifold;Submersion

This book gives a self-contained fundamental study of the subject. Besides the following special features it contains the author's detailed solution to the long-standing unsolved problem in the theory of complex manifolds: Does there exist a complex structure on the six-sphere? The special features of the book are: a classification of almost complex (and similarly, almost Hermitian) structures together with inclusion relations; discussions about various known almost Hermitian structures; a necessary and sufficient condition for a general almost Hermitian manifold to have constant holomorphic sectional (or bisectional) curvature and similar conditions for various special almost Hermitian manifolds; some complex Laplacians together with some of their relationships with the real Laplacian; the spectral geometry of Riemannian manifolds and some general almost Hermitian manifolds including K„hlerian manifolds as a special case; conditions for an almost complex structure to be a complex structure; some vanishing theorems for Riemannian and almost Hermitian manifolds.

This volume presents papers dedicated to Professor Shoshichi Kobayashi, commemorating the occasion of his sixtieth birthday on January 4, 1992.The principal theme of this volume is “Geometry and Analysis on Complex Manifolds”. It emphasizes the wide mathematical influence that Professor Kobayashi has on areas ranging from differential geometry to complex analysis and algebraic geometry. It covers various materials including holomorphic vector bundles on complex manifolds, Kähler metrics and Einstein-Hermitian metrics, geometric function theory in several complex variables, and symplectic or non-Kähler geometry on complex manifolds. These are areas in which Professor Kobayashi has made strong impact and is continuing to make many deep invaluable contributions.

The complex geometry deals with complex manifold which is a core of contemporary research in differential geometry. Over the last few decades, work in the field of complex manifold was slow. A. Gray worked on the basic structures of complex manifolds, such as Hermitian manifolds, nearly Kaehler manifolds, etc. S. Ishihara studied quaternion Kaehlerian manifolds. A. Bejancu generalized this notion in a very good manner and studied Geometry of CR-submanifolds. But B.Y. Chen in 1990, generalized the concept and established Geometry of slant submanifolds. Afterwards, several geometers tried to work on the generalized structure of complex manifolds. In this book, it has been tried to generalize the concept on complex manifolds. The initial chapters establish inequalities between the Ricci curvature and the squared mean curvature, B.Y. Chen inequalities for certain slant submanifolds and also establishes an inequality between the warping function and the squared mean curvature for totally real warped product submanifolds. This book tells about the study of certain structures of slant, semi-slant and bi-slant submanifolds in generalized complex space forms and in quaternion space forms.