Complex manifolds and deformation of complex structures

Author(s)

Bibliographic Information

Complex manifolds and deformation of complex structures

Kunihiko Kodaira ; translated by Kazuo Akao

（Die Grundlehren der mathematischen Wissenschaften, 283）

Springer-Verlag, c1986

  • : U.S.
  • : Germany
  • : pbk

Other Title

Fukuso tayōtairon

Available at  / 101 libraries

Note

Bibliography: p.459-460

Includes index

Description and Table of Contents

Volume

: U.S. ISBN 9780387961880

Table of Contents

1 Holomorphic Functions.- 1.1. Holomorphic Functions.- 1.2. Holomorphic Map.- 2 Complex Manifolds.- 2.1. Complex Manifolds.- 2.2. Compact Complex Manifolds.- 2.3. Complex Analytic Family.- 3 Differential Forms, Vector Bundles, Sheaves.- 3.1. Differential Forms.- 3.2. Vector Bundles.- 3.3. Sheaves and Cohomology.- 3.4. de Rham's Theorem and Dolbeault's Theorem.- 3.5. Harmonic Differential Forms.- 3.6. Complex Line Bundles.- 4 Infinitesimal Deformation.- 4.1. Differentiable Family.- 4.2. Infinitesimal Deformation.- 5 Theorem of Existence.- 5.1. Obstructions.- 5.2. Number of Moduli.- 5.3. Theorem of Existence.- 6 Theorem of Completeness.- 6.1. Theorem of Completeness.- 6.2. Number of Moduli.- 6.3. Later Developments.- 7 Theorem of Stability.- 7.1. Differentiable Family of Strongly Elliptic Differential Operators.- 7.2. Differentiable Family of Compact Complex Manifolds.- Appendix Elliptic Partial Differential Operators on a Manifold.- 1. Distributions on a Torus.- 2. Elliptic Partial Differential Operators on a Torus.- 3. Function Space of Sections of a Vector Bundle.- 4. Elliptic Linear Partial Differential Operators.- 5. The Existence of Weak Solutions of a Strongly Elliptic Partial Differential Equation.- 6. Regularity of Weak Solutions of Elliptic Linear Partial Differential Equations.
Volume

: pbk ISBN 9781461385929

Description

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).

Table of Contents

1 Holomorphic Functions.- 1.1. Holomorphic Functions.- 1.2. Holomorphic Map.- 2 Complex Manifolds.- 2.1. Complex Manifolds.- 2.2. Compact Complex Manifolds.- 2.3. Complex Analytic Family.- 3 Differential Forms, Vector Bundles, Sheaves.- 3.1. Differential Forms.- 3.2. Vector Bundles.- 3.3. Sheaves and Cohomology.- 3.4. de Rham's Theorem and Dolbeault's Theorem.- 3.5. Harmonic Differential Forms.- 3.6. Complex Line Bundles.- 4 Infinitesimal Deformation.- 4.1. Differentiable Family.- 4.2. Infinitesimal Deformation.- 5 Theorem of Existence.- 5.1. Obstructions.- 5.2. Number of Moduli.- 5.3. Theorem of Existence.- 6 Theorem of Completeness.- 6.1. Theorem of Completeness.- 6.2. Number of Moduli.- 6.3. Later Developments.- 7 Theorem of Stability.- 7.1. Differentiable Family of Strongly Elliptic Differential Operators.- 7.2. Differentiable Family of Compact Complex Manifolds.- Appendix Elliptic Partial Differential Operators on a Manifold.- 1. Distributions on a Torus.- 2. Elliptic Partial Differential Operators on a Torus.- 3. Function Space of Sections of a Vector Bundle.- 4. Elliptic Linear Partial Differential Operators.- 5. The Existence of Weak Solutions of a Strongly Elliptic Partial Differential Equation.- 6. Regularity of Weak Solutions of Elliptic Linear Partial Differential Equations.

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