復幾何導論

內(nèi)容概要

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 subject's 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.

書籍目錄

1  Local Theory　1.1  Holomorphic Functions of Several Variables　1.2  Complex and Hermitian Structures　1.3  Differential Forms2   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 Manifolds3  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 Metrics5 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-AlgebrasA  Hodge Theory on Differentiable ManifoldsB  Sheaf CohomologyReferencesIndex

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