Complex Differential Geometry AMS IP Studies in Advanced Mathematics 18 1st Edition by Fangyang Zheng ISBN 9780821829608 0821829602

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Product details: 

ISBN 10:  0821829602

ISBN 13:  9780821829608

Author:  Fangyang Zheng

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.

Table of contents: 

Part 1. Riemannian Geometry

Chapter 1. Differentiable Manifolds and Vector Bundles

1.1. Differentiable Manifolds

1.2. Tangent Spaces and Vector Fields

1.3. Vector Bundles

1.4. Tangent Bundles and Tensor Fields

1.5. The Topology of Smooth Manifolds

1.6. Lie Groups and Lie Algebras

Appendix: Topology, Homotopy and Covering Spaces Exercises

Chapter 2. Metric, Connection, and Curvature

2.1. Metric, Connection, and Curvature

2.2. Linear Connections and Geodesics

2.3. Riemannian Metrics and Riemannian Connections

2.4. Sectional, Ricci and Scalar Curvatures

2.5. Cartan’s Structure Equations and Examples

Exercises

Chapter 3. The Geometry of Complete Riemannian Manifolds

3.1. Riemannian Distance

3.2. Completeness and Hopf-Rinow Theorem

3.3. Jacobi Fields and Conjugate Points

3.4. Cartan-Ambrose-Hicks Theorem and Space Forms

3.5. Homogeneous and Symmetric Spaces

3.6. Hodge Theorem and Comparison Theorems

Exercises

Part 2. Complex Manifolds

Chapter 4. Complex manifolds and Analytic Varieties

4.1. Holomorphic Functions of One or More Complex Variables

4.2. Definition and Examples of Complex Manifolds

4.3. The Almost Complex Structure

4.4. More Examples

4.5. Hypersurfaces and Analytic Subvarieties

4.6. Divisors and Analytic Cycles

Chapter 5. Holomorphic Vector Bundles, Sheaves and Cohomology

5.1. Holomorphic Vector Bundles

5.2. Sheaves

5.3. Sheaf Cohomology Groups

5.4. Holomorphic Line Bundles

5.5. Chern Classes

Exercises

Chapter 6. Compact Complex Surfaces

6.1. The Topological Invariants

6.2. The Kodaira Dimension and the Algebraic Dimension

6.3. Examples of Surfaces

6.4. Enriques-Kodaira Classification Theory for Surfaces

Exercises

Part 3. Kähler Geometry

Chapter 7. Hermitian and Kähler Metrics

7.1. Connections on Vector Bundles and Their Curvature

7.2. Chern Forms of a Complex Vector Bundle

7.3. Hermitian Bundles

7.4. Hermitian and Kähler Metrics on Complex Manifolds

7.5. The Curvature of a Hermitian or Kähler Metric

7.6. Wu’s Theorem, Schwarz Lemma and Hartogs Phenomenon

Exercises

Chapter 8. Compact Kähler Manifolds

8.1. Hodge Theorem and Hodge Decomposition

8.2. The Hard Lefschetz Theorem

8.3. Kodaira Vanishing and Embedding Theorems

8.4. Ample Subvarieties and Ample Vector Bundles

8.5. Hermitian Symmetric Spaces and Kähler C-Spaces

8.6. The Hartshorne-Frankel Conjecture

Exercises

Chapter 9. Kähler Geometry

9.1. Calabi’s Conjecture and Kähler-Einstein Metrics

9.2. Corollaries of Yau’s Theorems

9.3. Invariant Metrics

9.4. Harmonic Maps and the Rigidity Theorems

9.5. Non-positively Curved Kähler Surfaces

Exercises

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Tags: Fangyang Zheng, Complex Differential, AMS IP Studies, Advanced Mathematics