基础代数几何（第2卷-第2版）（英文版）

作者简介

Books 2 and 3 correspond to Chap. V-IX of the first edition. They study schemes and complex manifolds, two notions that generalise in different directions the varieties in projective space studied in Book 1. Introducing them leads also to new results in the theory of projective varieties. For example, it is within the framework of the theory of schemes and abstract varieties that we find the natural proof of the adjunction formula for the genus of a curve, which we have already stated and applied in Chap. IV, 2.3. The theory of complex analytic manifolds leads to the study of the topology of projective varieties over the field of complex numbers. For some questions it is only here that the natural and historical logic of the subject can be reasserted; for example, differential forms were constructed in order to be integrated, a process which only makes sense for varieties over the (mai or) complex fields. Changes from the First Edition

书籍目录

BOOK 2. Schemes and Varieties

　Chapter V. Schemes

　　1. The Spec of a Ring

　　2. Sheaves

　　3. Schemes

　　4. Products of Schemes

　Chapter VI. Varieties

　　1. Definitions and Examples

　　2 Abstract and Quasiprojective Varieties

　　3 Coherent Sheaves

　　4 Classification of Geometric Objects and Universal Schemes

BOOK 3. Complex Algebraic Varieties and Complex Manifolds

　Chapter VII. The Topology of Algebraic Varieties

　　1. The Complex Topology

　　2. Connectedness

　　3. The Topology of Algebraic Curves

　　4. Real Algebraic Curves

　Chapter VIII. Complex Manifolds

　　1. Definitions and Examples

　　2. Divisors and Meromorphic Functions

　　3 Algebraic Varieties and Complex Manifolds

　　4. Kahler Manifolds

　Chapter IX. Uniformisation

　　1. The Universal Cover

　　2 Curves of Parabolic Type

　　3 Curves of Hyperbolic Type

　　4. Uniformising Higher Dimensional Varieties

Historical Sketch

　1. Elliptic Integrals

　2. Elliptic Functions

　3. Abelian Integrals

　4. Riemann Surfaces

　5. The Inversion of Abelian Integrals

　6. The Geometry of Algebraic Curves

　7. Higher Dimensional Geometry

　8. The Analytic Theory of Complex Manifolds

　9. Algebraic Varieties over Arbitrary Fields and Schemes

References

Index

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