代數(shù)幾何I

前言

要使我國的數(shù)學(xué)事業(yè)更好地發(fā)展起來，需要數(shù)學(xué)家淡泊名利并付出更艱苦地努力。另一方面，我們也要從客觀上為數(shù)學(xué)家創(chuàng)造更有利的發(fā)展數(shù)學(xué)事業(yè)的外部環(huán)境，這主要是加強對數(shù)學(xué)事業(yè)的支持與投資力度，使數(shù)學(xué)家有較好的工作與生活條件，其中也包括改善與加強數(shù)學(xué)的出版工作。從出版方面來講，除了較好較快地出版我們自己的成果外，引進國外的先進出版物無疑也是十分重要與必不可少的。從數(shù)學(xué)來說，施普林格（springer）出版社至今仍然是世界上最具權(quán)威的出版社?？茖W(xué)出版社影印一批他們出版的好的新書，使我國廣大數(shù)學(xué)家能以較低的價格購買，特別是在邊遠地區(qū)工作的數(shù)學(xué)家能普遍見到這些書，無疑是對推動我國數(shù)學(xué)的科研與教學(xué)十分有益的事。這次科學(xué)出版社購買了版權(quán)，一次影印了23本施普林格出版社出版的數(shù)學(xué)書，就是一件好事，也是值得繼續(xù)做下去的事情。大體上分一下，這23本書中，包括基礎(chǔ)數(shù)學(xué)書5本，應(yīng)用數(shù)學(xué)書6本與計算數(shù)學(xué)書12本，其中有些書也具有交叉性質(zhì)。這些書都是很新的，2000年以后出版的占絕大部分，共計16本，其余的也是1990年以后出版的。這些書可以使讀者較快地了解數(shù)學(xué)某方面的前沿，例如基礎(chǔ)數(shù)學(xué)中的數(shù)論、代數(shù)與拓撲三本，都是由該領(lǐng)域大數(shù)學(xué)家編著的“數(shù)學(xué)百科全書”的分冊。對從事這方面研究的數(shù)學(xué)家了解該領(lǐng)域的前沿與全貌很有幫助。按照學(xué)科的特點，基礎(chǔ)數(shù)學(xué)類的書以“經(jīng)典”為主，應(yīng)用和計算數(shù)學(xué)類的書以“前沿”為主。這些書的作者多數(shù)是國際知名的大數(shù)學(xué)家，例如《拓撲學(xué)》一書的作者諾維科夫是俄羅斯科學(xué)院的院士，曾獲“菲爾茲獎”和“沃爾夫數(shù)學(xué)獎”。這些大數(shù)學(xué)家的著作無疑將會對我國的科研人員起到非常好的指導(dǎo)作用。當然，23本書只能涵蓋數(shù)學(xué)的一部分，所以，這項工作還應(yīng)該繼續(xù)做下去。更進一步，有些讀者面較廣的好書還應(yīng)該翻譯成中文出版，使之有更大的讀者群?？傊覍茖W(xué)出版社影印施普林格出版社的部分數(shù)學(xué)著作這一舉措表示熱烈的支持，并盼望這一工作取得更大的成績。

內(nèi)容概要

This book consists of two parts. The first is devoted to the theory of curves, which are treated from both the analytic and algebraic points of view. Starting with the basic notions of the theory of Riemann surfaces the reader is lead into an exposition covering the Riemann-Roch theorem, Riemann's fundamental existence theorem.uniformization and automorphic functions. The algebraic material also treats algebraic curves over an arbitrary field and the connection between algebraic curves and Abelian varieties. The second part is an introduction to higher- dimensional algebraic geometry. The author deals with algebraic varieties, the corresponding morphisms,the theory of coherent sheaves and, finally, The theory of schemes.This book is a very readable introduction to algebraic geometry and will be immensely useful to mathematicians working in algebraic geometry and complex analysis and especially to graduate students in these fields.

作者簡介

作者：(俄羅斯)沙法列維奇 (I.R.Shafarevich)

書籍目錄

Introduction by I.R.Shafaxevich  Chapter 1.Riemann Surfaces    §1.Basic Notions      1.1.Complex Chart；Complex Coordinates      1.2.Complex Analytic Atlas      1.3.Complex Analytic Manifolds      1.4.Mappings of Complex Manifolds      1.5.Dimension of a Complex Manifold    1.6.Riemann Surfaces      1.7.Di6erentiable Manifolds    §2.Mappings of Riemann Surfaces      2.1.Nonconstant Mappings of Riemann Surfaces axe Discrete      2.2.Meromorphic Functions on a Pdemann Surface      2.3.Meromorphic Functions With Prescribed Behaviour at Poles    2.4.Multiplicity of a Mapping；Order of a Function      2.5.Topological Properties of Mappings of Riemann Surfaces      2.6.Divisors on Riemann Surfaces      2.7.Finite Mappings of Riemann Surfaces      2.8.Unramified Coverings of Pdemann Surfaces      2.9.The Universal Covering      2.10.COntinuation of Mappings      2.n.The Riemann Surface of al2 Algebraic Function    §3.Topology of Riemann Surfaces      3.1.Orientability    3.2.Triangulability    3.3.Development；Topological Genus      3.4.Structure of the Fundamental Group      3.5.The Euler Characteristic      3.6.The Hurwitz Formulae      3.7.Homology and Cohomology；Betti Numbers    3.8.Intersection Product；PoincareDUalitV    §4.Calculus on Riemann Surfaces    4.1.Tangent Vectors；Differentiations      4.2.Differential Forms      4.3.Exterior Differentiations；de Rham Cohomology    4.4.Kihler and Riemann Metrics      4.5.Integration of Exterior Differentials；Gteen，s Formula      4.6.Periods；Rham Isomorphism      4.7.Holomorphic Diffentials；Geometric Genus；Riemann，S Bilinear Delations      4.8.Meromorphic Differentials；Canonical Divisors      4.9.Meromorphic Differentials with Prescribed Behaviour at P0les；Residues      4.10.Periods of Meromorphic Differentials      4.11.Harmonic Differentials    4.12.Hilbert Space of Differentials；Harmonic Projection      4.13.Hodge Decomposition      4.14.Existence of Meromorphic Differentials and Functions    4.15.Dirichlet’S Principle    §5.Classification of njemann Surfaces      5.1.Canonical Regions    5.2.Uniformization      5.3.Types of Riemann Surfaces      5.4.Automorphisms ofCanonical Regions      5.5.Pdemann Surfaces of Elliptic Type      5.6.Riemann Surfaces of Parabolic Type      5.7.Riemann Surfaces ofHyperbolic Type    5.8.Automorphic Forms；Poincar6 Series      5.9.Quotient Riemann Surfaces；the Absolute Invariant      5.10.Moduli of Riemann Surfaces  §6.Algebraic Nature of Compact Riemann Surfaces      6.1.Function Spaces and Mappings Associated with Divisors    6.2.Riemann.RDch Formula；Reciprocity Law for Differentialsof the First and Second Kind      6.3.Applications of the Riemann—nDch Formula to Problems0f Existence of Meromorphic Functions and Differentials    6.4.Compact Riemann Surfaces are Projective      6.5.Algebraic Nature of Projective Models；Arithmetic Riemann Surfaces      6.6.Models of Riemann Surfaces of Genus lChapter 2.Algebraic Curves  Chapter 3.Jaclbians and Abelian Varieties References

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