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內容簡介

Complex geometry is a highly attractive branch of modern mathematics that has witnessed many years of active and successful research and that has re- cently obtained new impetus from physicists interest in questions related to mirror symmetry. Due to its interactions with various other fields (differential, algebraic, and arithmetic geometry, but also string theory and conformal field theory), it has become an area with many facets. Also, there are a number of challenging open problems which contribute to the subjects attraction. The most famous among them is the Hodge conjecture, one of the seven one-million dollar millennium problems of the Clay Mathematics Institute. So, it seems likely t at this area will fascinate new generations for many years to come.

內頁插圖

目錄

1 Local Theory
1.1 Holomorphic Functions of Several Variables
1.2 Complex and Hermitian Structures
1.3 Differential Forms
2 Complex Manifolds
2.1 Complex Manifolds: Definition and Examples
2.2 Holomorphic Vector Bundles
2.3 Divisors and Line Bundles
2.4 The Projective Space
2.5 Blow-ups
2.6 Differential Calculus on Complex Manifolds
3 Kahler Manifolds
3.1 Kahler Identities
3.2 Hodge Theory on Kahler Manifolds
3.3 Lefschetz Theorems
Appendix
3.A Formality of Compact Kahler Manifolds
3.B SUSY for Kahler Manifolds
3.C Hodge Structures
4 Vector Bundles
4.1 Hermitian Vector Bundles and Serre Duality
4.2 Connections
4.3 Curvature
4.4 Chern Classes
Appendix
4.A Levi-Civita Connection and Holonomy on Complex Manifolds
4.B Hermite-Einstein and Kahler-Einstein Metrics
5 Applications of Cohomology
5.1 Hirzebruch-Riemann-Roch Theorem
5.2 Kodaira Vanishing Theorem and Applications
5.3 Kodaira Embedding Theorem
Deformations of Complex Structures
6.1 The Maurer-Cartan Equation
6.2 General Results
Appendix
6.A dGBV-Algebras
A Hodge Theory on Differentiable Manifolds
B Sheaf Cohomology
References
Index

前言/序言

　　Complex geometry is a highly attractive branch of modern mathematics thathas witnessed many years of active and successful research and that has recently obtained new impetus from physicists interest in questions related tomirror symmetry. Due to its interactions with various other fields （differential，algebraic， and arithmetic geometry， but also string theory and conformal fieldtheory）， it has become an area with many facets. Also， there are a number ofchallenging open problems which contribute to the subjects attraction. Themost famous among them is the Hodge conjecture， one of the seven one-milliondollar millennium problems of the Clay Mathematics Institute. So， it seemslikely that this area will fascinate new generations for many years to come.
　　Complex geometry， as presented in this book， studies the geometry of（mostly compact） complex manifolds. A complex manifold is a differentiablemanifold endowed with the additional datum of a complex structure which ismuch more rigid than the geometrical structures in differential geometry. Dueto this rigidity， one is often able to describe the geometry of complex manifoldsin very explicit terms. E.g. the important class of projective manifolds can， inprinciple， be described as zero sets of polynomials.
　　Yet， a complete classification of all compact complex manifolds is toomuch to be hoped for. Complex curves can be classified in some sense （in-volving moduli spaces etc.）， but already the classification of complex surfacesis tremendously complicated and partly incomplete.
　　In this book we will concentrate on more restrictive types of complexmanifolds for which a rather complete theory is in store and which are alsorelevant in the applications. A prominent example are Calabi-Yau manifolds，which play a central role in questions related to mirror symmetry. Often，interesting complex manifolds are distinguished by the presence of specialRiemannian metrics. This will be one of the central themes throughout thistext. The idea is to study cases where the Riemannian and complex geometryon a differentiable manifold are not totally unrelated.

復幾何導論（英文版） [Complex Geomety:An Introduction] 下載 mobi epub pdf txt 電子書

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