本书内容为原作者版权所有，未经协议授权，禁止下载使用
主 编 杨劲根
副主编 楼红卫 李振钱 郝群
编写人员（按汉语拼音为序）
陈超群 陈猛 东瑜昕 高威 郝群 刘东弟 吕志 童裕孙 王巨平 王泽军 徐晓津 杨劲根 应坚刚 张锦豪 张永前 周子翔 朱胜林
1. 序言
2. 非数学专业的数学教材
3.数学分析和泛函分析
4.单复变函数
5.多复变函数
6.代数
7.数论
8.代数几何
9.拓扑与微分几何
10.偏微分方程
11.概率论
12.计算数学
13.其他
14.附录
代数几何是核心数学的重要分支，内容比较高深，不太容易入门。由于它所用的知识比较多，学习的周期相对比较长。一般本科生阶段不设代数几何课程。
代数几何对抽象代数、复分析、拓扑等都有较高的要求，特别交换代数和同调代数是它的不可缺少的工具。我们在这里介绍一些目前在国际上使用最多的一批基础性的教材供有兴趣和有志向的读者参考。
书名：An Invitation to Algebraic Geometry
作者：K.Smith etc.
出版商：Springer-Verlag, New York, 2000. ISBN 0-387-98980-3
页数：155
适用范围：基础数学本科高年级或非代数几何专业研究生低年级
预备知识：线性代数，群，环，域扩张，Galois
理论的基础知识，最基本的点集拓扑
习题数量：大
习题难度：容易
推荐强度：8.5
书评： 本书由作者 1996 年的为非代数几何专业的数学研究生开设的 20 小时代数几何课的讲义 整理修改而成，是一本非常好的代数几何入门书。
本书从最基本的交换代数代数几何概念开始，用简明的方式引入仿射簇和射影簇。重点的内容是一些经典的例子： Veronese 映射、计数几何、Segr\'{e} 嵌入、 Grassman 簇等。最后介绍代数几何中的一些重大问题如奇点解消、射影簇分类、典范映射等。
本书把代数几何讲解的具体易懂，不拘泥于细节，一些关键定理给出清晰的解释而不是详细的证明，如 Hilbert 零点定理、 B\'{e}zout 定理、 Bertini 定理等，有一部分内容的简单证明放入习题。有些其他教材不提及的问题也作了简短介绍，如 Gauss 映射用来 2 页和 5 道习题。书中不时插入对一些重要问题研究的历史和现状，颇有 Wikipedia 的风格。在三个合适的地方叙述三个未解决的难题： Jabobian 猜想、空间曲线的完全交问题、 Iitaka 猜想。
本书篇幅不大，适合初学者在较短的时间内对代数几何的特点有初步的了解，为进一步的深入学习作准备。（杨劲根）
国外评论摘选
1）This book has a great deal to recommend it:
a. It is a genuinely entry-level book that begins with the definition of a prime ideal and the Nullstellensatz.
b. The style of explanation is clearly geared to noninsiders. In addition to giving examples of algebraic varieties, some nonexamples are given that might have occurred to, say, an analyst as reasonable objects to study but that do not qualify as varieties.
c. The illustrations are frequent, relevant, and well executed.
d. The authors go out of their way to help the reader develop geometric intuition and to relate it to the accompanying algebraic description. For example, in the careful treatment of the geometry of a family of hyperbolas in chapter 6, the geometry of the general hyperplane section is beautifully illustrated, and the reader's geometric intuition is stimulated into action.
e. Many of the constructions covered are classical-the Grassmannian, the Veronese, the Gauss mapping, the secant variety of a variety-yet the book almost seamlessly connects this with more modern material, such as resolution of singularities and vector bundles.
f. There is a consistent policy throughout the book of tying in elementary algebraic geometry to recent developments by current leaders such as Kollar, Kontsevich,Mori, Lazarsfeld, and de Jong, so that readers come away with a clear conception of where this is all going and what the next steps might be if a particular topic sparks their interest. Overall, readers will find this book easy to get into and enjoyable to read. Outsiders to the subject will feel that they are hiking up a gently sloping trail, at the end of which they reach a number of pleasant viewing spots from which they. can see rather far in a number of different directions. Students contemplating algebraic geometry as a field of specialization will also find this an attractive and instructive place to start. ( by Mark Green, {\em The American Mathematical Monthly,} Vol. {\bf 109}, 675-678(2002))
2）This could be your only book on algebraic geometry if you just want a sound idea of what algebraic geometry can do. If you actually want to know the field, and you do not already have a lot of expert friends telling you about it, then the advanced books will go much more easily with this expert around. It is a terrific guide to the key ideas--what they mean, how they work, how they look. The only book like this one in brevity and scope is Reid UNDERGRADUATE ALGEBRAIC GEOMETRY--with its highly informed, highly polemical, final chapter on the state of the art. Both are very good. This one is more advanced. Beyond what Reid covers, Smith sketches Hilbert polynomials, Hironaka's (and very briefly even De Jong's) approach to removing singularities, and ample line bundles. You do need a bit of topology and analysis to follow it. Smith has very many fewer concrete examples than Reid. They are beautifully chosen classics, like Veronese maps and Segre maps, so they teach a lot. And the more you know to start with, the more you will see in each.
The book does geometry over the complex numbers. It is good old conservative material, with terrific graphics of curves and surfaces. The proofs and partial proofs are very clear, intuitive and to the point. But, in fact, just because the proofs are so clear and to the point they usually work in a much broader setting. Long stretches of the book apply just as well over any field or any algebraically complete field. This generality is only mentioned a few times, in passing, but is there if you want it. Smith describes schemes very briefly, and mentions them at each point where they naturally arise. You will not know what schemes are at the end of this book. You will know some things they DO. She has no time for fights between concretely complex and abstractly scheming approaches--for her it is all geometry.
3）For people just starting on Algebraic Geometry, Robin Hartshorne's book, is very daunting--but it is the ULTIMATE book for professional and advanced readers. But for starters, Karen Smith's An Invitation to Algebraic Geometry is simply a SPLENDID way to start working on the basic ideas. The author has some stunning graphs and pictures to help understand material. I loved the book the minute I opened it.
书名：Introduction to Commutative Algebra and Algebraic Geometry
作者： Ernst Kunz
出版商： Birkhauser Boston, (1985) ISBN 3-7643-3065-1
页数：238
适用范围：基础数学本科高年级或研究生低年级
预备知识：线性代数，群，环，域扩张，Galois 理论的基础知识，最基本的点集拓扑
习题数量：大
习题难度： 大部分中等，少量难题
推荐强度：9
书评： 作为交换代数的入门书，它不如 Atiyah-McDonald 的有名，也不如 Eisenbud 的大， 但是我认为对于有志学习代数几何的大学生来讲，这是最好的入门书。此书与大学基础课程 的衔接非常紧密，不管是自学还是用此书当教材都比较轻松，如果再认真做习题则效果更好。 交换代数的内容甚广，作者完全按经典代数几何的需要选择交换代数的内容，重点是多变量 多项式环的商环及它上的有限生成模。除了基础性的材料外，也有少量研究性的题材。现在 代数几何最流行的研究生教材是 Hartshorne 的 Algebraic Geometry, 本书可以认为是 Hartshorne 的教程的前续课程。本书作者是著名的交换代数专家，原书用德文写作，后翻译成英文，美国 代数几何大师 David Mumford 写了序言，称此书是美国学代数几何学生久等的一本书，它填补 了一个空白。（杨劲根）
书名：Basic Algebraic Geometry (Second, Revised and Expanded Edition)
作者： Shafarevich
出版商： Springer-Verlag (1988) ISBN 3-540-54812-2
页数：上册 303 下册 269
适用范围：基础数学研究生
预备知识：近世代数、复分析、点集拓扑
习题数量：大
习题难度：较难
推荐强度：9
书评： 本书是俄罗斯的数学大师 Shafarevich 的力作，由英国著名代数几何学家 Miles Reid 翻译成英文。本书内容非常丰富且不枯燥，叙述和证明清晰，比较容易读，是非常收欢迎的一本 代数几何书，国内外不少院校开设代数几何课曾将此书选为研究生教材。全书分三大部分， 第一部分是射影簇，内容包含经典代数几何，一直讲到代数曲面的分类和奇点，其中不乏其它 教科书中不多见的内容。这一部分占了整个上册，作为一个学期的课程内容够多的。 第二部分是概形理论，用现代的语言来刻画代数簇，最后讲到 Hilbert 概形。本部分内容比较简要，基本上讲清概形和层论的威力。第三部分是复代数流形的拓扑和几何，很多内容如代数簇的拓扑 分类和 uniformization 在其它代数几何教科书很难找到。总之，本书基本上讲述了代数几何的所有方法。 习题非常丰富，大部分的习题很有意思，可以看出是作者和他的助手们多年积累而编成的。 还有一个显著的特点是本书不需要交换代数的预备知识，当然学过交换代数在看此书更加轻松。 （杨劲根）
国外评论摘选
1) I have been a student of AG for the past six years and I have come to the conclusion that Shafarevich is a great place to start. Having said this, one must have the necessary background in algebra and topology. I disagree with the other reviewer about doing this after Hartshorne--start here then do Hartshorne!!!
书名：Algebraic Geometry
作者: Robin Hartshorne
出版商：Springer-Verlag
页数：495
适用范围：基础数学研究生
预备知识：近世代数、交换代数、同调代数、复分析、基础拓扑
习题数量：多
习题难度： 又难又繁
推荐强度：8.8
书评： 本书是现代代数几何的标准教科书，适合代数几何专业的研究生使用。从代数几何的发展历史来看， 60年代由 Grothendieck 提出的以概形为基础的新理论完成了代数几何的一次新的革命，至今代数几何仍以 Grothendieck 的理论为基础，他和 Dieudonne 合写的庞大的 EGA (Elements de Geometrie Algebrique) 可堪为代数几何的圣经。但是由于规模太大，EGA 无法当作教科书使用。事实上在 EGA 之前，Grothendieck 在日本的东北数学杂志上一篇同调代数的长文也是代数几何的奠基性的文献之一。此后，出现了两本有 很大影响的书，其一是 Matsumura 的 Commutative Algebra, 这本看上去象是研究笔记的专著把 Grothendieck 的 EGA 中的交换代数部分和部分同调代数整理出来加以详细证明。另一本就是 Hartshorne 的 Algebraic Geometry, 此书用两章约230页的篇幅介绍 Grothendieck 的概形理论。作者能完成此举得益于两点：第一，所有和交换代数 有关的内容都引用 Matsumura 的有关章节。第二，作者牺牲一般性而大大简化了很多大定理的证明，具体来讲， 在大部分章节作者把概形限制为诺特概形，把态射限制为有限型的态射。这样的简化对代数几何的主流方向的研究 来说影响不大。从某种意义来看，Hartshorne 的书是 Grothendieck 的 EGA 的浓缩简化版。这本书中的很多英文 名词现在已经获得代数几何界的普遍认可。
本书中定理的证明比较简洁，认真的读者需要补充不少细节，从这点来看，此书的浓度比较大，要读懂此书大部分得化一年以上的 时间，对于没有学过交换代数或学的不多的读者，最好先读 Kunz 的 Introduction to Commutative Algebra and Algebraic Geometry 一书。会法文的读者可以参考 Grothendieck 的 EGA 学习，因为有些定理的证明在 EGA 中写的更为详细。 习题也是本书的一大难点，不少研究生抱怨这本书中的习题太难做。网上有些地方甚至有 Hartshorne 习题部分解答下载。 （杨劲根）
国外评论摘选
1) This book is one of the most used in graduate courses in algebraic geometry and one that causes most beginning students the most trouble. But it is a subject that is now a must-learn for those interested in its many applications, such as cryptography, coding theory, physics, computer graphics, and engineering. That algebraic geometry has so many applications is quite amazing, since it was not too long ago that it was thought of as a highly abstract, esoteric topic. That being said, most of the books on the subject, including this one, are written from a very formal point of view. Those interested in applications will have to face up to this when attempting to learn the subject. To read this book productively one should gain a thorough knowledge of commutative algebra, a good start being Eisenbud's book on this subject. Also, it is important to dig into the original literature on algebraic geometry, with the goal of gaining insight into the constructions and problems involved. The author of this book does not make an attempt to motivate the subject with historical examples, and so such a perusal of the literature is mandatory for a deeper appreciation of algebraic geometry. The study of algebraic geometry is well worth the time however, since it is one that is marked by brilliant developments, and one that will no doubt find even more applications in this century. Varieties, both affine and projective, are introduced in chapter 1. The discussion is purely formal, with the examples given unfortunately in the exercises. The Zariski topology is introduced by first defining algebraic sets, which are zero sets of collections of polynomials. The algebraic sets are closed under intersection and under finite unions. Therefore their complements form a topology which is the Zariski topology. The properties of varieties are discussed, along with morphisms between them. Functionals on varieties, called regular functions in algebraic geometry, are introduced to define these morphisms. Rational and birational maps, so important in classical algebraic geometry are introduced here also. Blowing up is discussed as an example of a birational map. A very interesting way, due to Zariski, of defining a nonsingular variety intrinsically in terms of local rings is given. The more specialized case of nonsingular curves is treated, and the reader gets a small taste of elliptic curves in the exercises. A very condensed treatment of intersection theory in projective space is given. The discussion is primarily from an algebraic point of view. It would have been nice if the author would have given more motivation of why graded modules are necessary in the definition of intersection multiplicity.
The theory of schemes follows in chapter 2, and to that end sheaf theory is developed very quickly and with no motivation (such as could be obtained from a discussion of analytic continuation in complex analysis). Needless to say scheme theory is very abstract and requires much dedication on the reader's part to gain an in-depth understanding. I have found the best way to learn this material is via many examples: try to experiment and invent some of your own. The author's discussion on divisors in this chapter is fairly concrete however.
The reader is introduced to the cohomology of sheaves in chapter 3, and the reader should review a book on homological algebra before taking on this chapter. Derived functors are used to construct sheaf cohomology which is then applied to a Noetherian affine scheme, and shown to be the same as the Cech cohomology for Noetherian separated schemes. A very detailed discussion is given of the Serre duality theorem.
Things get much more concrete in the next chapter on curves. After a short proof o the Riemann-Roch theorem, the author studies morphisms of curves via Hurwitz's theorem. The author then treats embeddings in projective space, and shows that any curve can be embedded in P(3), and that any curve can be mapped birationally into P(2) if one allows nodes as singularities in the image. And then the author treats the most fascinating objects in all of mathematics: elliptic curves. Although short, the author does a fine job of introducing most important results.
This is followed in the next chapter by a discussion of algebraic surfaces in the last chapter of the book. The treatment is again much more concrete than the earlier chapters of the book, and the author details modern formulations of classical constructions in algebraic geometry. Ruled surfaces, and nonsingular cubic surfaces in P(3) are discussed, as well as intersection theory. A short overview of the classification of surfaces is given. The reader interested in more of the details of algebraic surfaces should consult some of the early works on the subject, particularly ones dealing with Riemann surfaces. It was the study of algebraic functions of one variable that led to the introduction of Riemann surfaces, and the later to a consideration of algebraic functions of two variables. A perusal of the works of some of the Italian geometers could also be of benefit as it will give a greater appreciation of the methods of modern algebraic geometry to put their results on a rigorous foundation.
2) This is THE book to use if you're interested in learning algebraic geometry via the language of schemes. Certainly, this is a difficult book; even more so because many important results are left as exercises. But reading through this book and completing all the exercises will give you most of the background you need to get into the cutting edge of AG. This is exactly how my advisor prepares his students, and how his advisor prepared him, and it seems to work. Some helpful suggestions from my experience with this book: 1) if you want more concrete examples of schemes, take a look at Eisenbud and Harris, The Geometry of Schemes; 2) if you prefer a more analytic approach (via Riemann surfaces), Griffiths and Harris is worth checking out, though it lacks exercises.
书名：Principles of Algebraic Geometry
作者： Phillip Griffiths, Joseph Harris
出版商： John Wiley & Sons, Inc., 1978; ISBN 0-471-32792-1
页数：813
适用范围：基础数学研究生
预备知识：近世代数、复分析、点集拓扑
习题数量：无
推荐强度：8
书评： 代数几何原理的作者之一 P. Griffiths 是美国科学院院士，国际著名的数学家，他是 普林斯顿高等研究院教授。本书从比较解析的方面介绍代数几何，它给人的印象是代数几何也很具体， 这是一本代数几何的入门引导课本。
本书前两章处理复流形理论的一些结果和技巧，同时强调它们在射影代数簇上的应用； 第2章开始介绍黎曼面和代数曲线的理论；第3章中介绍了流、陈类、Riemann-Roch 公式等基本工具，然后在第4章中介绍代数曲面理论；本书最后介绍 Quadric Line Complex.
本书的选材十分合适，内容基本自我包容，给读者以直观和易懂的感觉， 但笔者认为该书在排版过程中过于仓促，书中（尤其第0章）累出打印错误和符号冲突。 如果有谁能从现在的版本重新修订并浓缩成一本稍薄的代数几何入门书，那将是代数几何爱好者的福音。（陈猛）
国外评论摘选
1) Once thought to be highly esoteric and useless by those interested in applications, algebraic geometry has literally taken the world by storm. Indeed, coding theory, cryptography, steganography, computer graphics, control theory, and artificial intelligence are just a few of the areas that are now making heavy use of algebraic geometry. This book would probably be one the most useful one for those interested in applications, for it is an overview of algebraic geometry from the complex analytic point of view, and complex analysis is a subject that most engineers and scientists have had to learn at some point in their careers. But one must not think that this book is entirely concrete in its content. There are many places where the authors discuss concepts that are very abstract, particularly the discussion of sheaf theory, and this might make its reading difficult. The complex analytic point of view however is the best way of learning the material from a practical point of view, and mastery of this book will pave the way for indulging oneself in its many applications.
Algebraic geometry is an exciting subject, but one must master some background material before beginning a study of it. This is done in the initial part of the book (Part 0), wherein the reader will find an overview of harmonic analysis (potential theory) and Kahler geometry in the context of compact complex manifolds. Readers first encountering Kahler geometry should just view it as a generalization of Euclidean geometry in a complex setting. Indeed, the so-called Kahler condition is nothing other than an approximation of the Euclidean metric to order 2 at each point.
The authors choose to introduce algebraic varieties in a projective space setting in chapter 1, i.e. they are the set of complex zeros of homogeneous polynomials in projective space. The absence of a global holomorphic function for a compact complex manifold motivates a study of meromorphic functions and divisors. Divisors are introduced as formal sums of irreducible analytic hypersurfaces, but they are related to the defining functions for these hypersurfaces also, via the poles and zeros of meromorphic functions. For the mathematical purist, a sheafified version of divisors is also outlined. Divisors and line bundles are basically linear tools used to investigate complex varieties through their representation as complex submanifolds of projective space. In addition, various approaches are used to study codimension-one subvarieties, such as the results of Kodaira and Spencer. Although the famous Kodaira vanishing theorem is clothed in the language of Cech cohomology, this cohomology is represented by harmonic forms, thus making its understanding more accessible. The authors also show explicitly to what extent an algebraic variety can be thought of as a compact complex manifold via the Kodaira embedding theorem. Projective space of course is not the most complicated of constructions, as readers familiar with the theory of vector bundles will know. Grassmannians are an example of this, and they are introduced and discussed in the book as generalizations of projective space. And, just as in the ordinary theory of vector bundles, the authors show how to use Grassmannians to act as universal bundles for holomorphic vector bundles.
The presence of meromorphic functions will alert the astute reader as to the role of Riemann surfaces in the study of complex algebraic varieties. Indeed, in chapter 2, the authors cast many classical complex analytic results to modern ones, and they prove the famous Riemann-Roch theorem, which essentially counts the number of meromorphic functions on a Riemann surface of genus g. The theory of Abelian varieties is outlined, and the reader gets a taste of Italian algebraic geometry but done in the rigorous setting of Plucker formulas and coordinates.
Chapter 3 is a summary of some of the other methodologies and techniques used to study general analytic varieties, the first of these being the theory of currents, i.e differential forms with distribution coefficients. It is perhaps not surprising to see this applied here, given that it can handle both the smooth and piecewise smooth chains simultaneously. The currents are associated to analytic varieties and allow a definition of their intersection numbers and a proof that they are positive. The all-important Chern classes are introduced here, and it is shown that the Chern classes of a holomorphic vector bundle over an algebraic variety are fundamental classes of algebraic cycles. Most importantly the authors introduce spectral sequences, a topic that is usually formidable for newcomers to algebraic geometry.
The study of surfaces is studied in chapter 4, with the differences between its study and the theory of curves (Riemann surfaces) emphasized. The reader gets a first crack at the notion of a rational map, and the birational classification of surfaces is shown. Intuitively, one expects that the classification of surfaces would be easy if it were not for singular points, and this is born out in the use of blowing up singularities in this chapter. Rational surfaces are characterized using Noether's lemma, and a rather detailed discussion is given of surfaces that are not rational, giving the reader more examples of rigorous Italian geometry.
2) If you are a graduate student in mathematics or related fields and you are interested in learning algebraic geometry in the Griffiths-Harris way, then I suggest before buying this book to have a good background in the following: 1. Complex Analysis 2. Differential Geometry and calculus on manifolds 3. Homology-Cohomology Theory 4. Undergraduate Algebraic Geometry
Do not expect chapter 0, Foundational Material, to be the place where you are supposed to build your foundation. You can try the books of Michael Spivak, David A. Cox, Fangyang Zheng, among other books for foundational material but not chapter 0.
However, if you have most of the above-mentioned foundational material, then this book is good in presenting complex manifolds for example in chapter 0 section 2 and also in presenting (complex) holomorphic vector bundles, as well as many other things.
So, in summary, I would say a good book but not for students trying to learn the basics in algebraic geometry.
书名：The Red Book of Varieties and Schemes
作者： David Mumford
出版商： Springer-Verlag (1994) ISBN 3-540-50497-4
页数：309
适用范围：基础数学研究生
预备知识：近世代数、复分析、点集拓扑
习题数量：很少
习题难度： 较难
推荐强度：8.6
书评： 代数几何学家 Mumford 是美国科学院院士，菲尔滋奖获得者，在哈佛大学任教多年。 本书实际上是 60 年代他在哈佛的代数几何课程的讲义。即使到80年代有了 Hartshorne, Shafarevich 写的优秀教科书，很多初学者仍然喜欢 Mumford 的老讲义,油印本在研究生中广为流传。Springer Verlag 经专家的推荐便将这些讲义原封不动地出版了，由于原来的油印的封面是红色的，故此书就被亲切地取名为 red book. 比较可惜的是这本书只有三章，由于种种原因作者未能把原来写讲义的庞大计划执行到底。
代数几何是抽象概念非常多的一门数学分支，初学者需要化很大的精力来理解、消化和记住一大堆基本概念。 Mumford 的讲义用朴实无华的方式解释代数几何这些概念的来龙去脉，一点不落俗套。从目录上看，这些内容和 其它同类的书差不多，事实上具体的论述还是很不一样的，讲义行文的非正式的风格也使枯燥的数学变的生动。 作者自嘲地称这本书里一个定理也没有，这多少有些夸张， 很多应该叫做定理的结论在这位大师面前大概不能称为定理。 （杨劲根）
国外评论摘选
1）In a nutshell, reading this book is like reading the mind of a great mathematician as he thinks about a great new idea. Anyone interested in schemes should read it. But a review needs more detail: The RED BOOK is a concise, brilliant survey of schemes, by one of the first mathematicians to learn of them from Grothendieck. He gives wonderfully intuitive pictures of schemes, especially of arithmetic schemes where number theory appears as geometry. The geometry shines through it all: as in differentials, and etale maps, and how unique factorization relates to non-singularity. There is a bravura discussion of Zariski's Main Theorem (the algebraic property of being normal implies that a variety has only one branch at each point) comparing forms of it from older algebraic geometry, topology, power series, and schemes. Mumford cites proofs of these but does not give them. In fact, this theorem was one of the first things Mumford could use, to get Zariski to respect schemes.
Many accomplished algebraic geometers say this book got them started. But you probably cannot learn to work in the subject from this book alone--you either have to work with people who work with it, or use some other books besides (maybe both). The other book would probably be Hartshorne ALGEBRAIC GEOMETRY, which is far more detailed, has far more examples, goes very much farther into cohomology--and is very much longer and denser (though also clearly written).
2）There is a problem in getting going with alg. geo. To learn the geometry you need commutative algebra and to contextualize commutative algebra you need algebraic geometry. Mumford is an excellent book to get going without the need for the heavy prereqs of the more classic books like Hartshorne or Griffiths-Harris. A really good read. This is not however a terrific reference text, you'll need something else as a reference. Its much to expository and their is no index.
书名：Compact Complex Surfaces, 2nd edition
作者：W.P. Barth, K. Hulek, C.A.M. Peters, A. Van de Ven
出版商：Springer-Verlag (2003) ISBN 3-540-00832-2
页数：436
适用范围：代数几何、复几何、微分几何方向研究生
预备知识：多复变函数论、代数几何、复微分几何
习题数量：无
推荐强度：9
书评： 紧复曲面专著第一版于1984年出版，自其出版以来因其选材精致，重点突出而广受青睐， 内行称其为BPV（第一版的三位作者的姓氏的开头字母）。第二版增加了不少新的研究成果， 但笔者认为第二版的组织过于仓促，反而给人画蛇添足的感觉，尽管如此， 这仍不失为一本优秀的专业工具书。
本书第一章列出了必需的预备知识，虽然没有证明，但笔者认为该内容十分恰当。 第二章中，作者分别介绍了曲面上的曲线、Riemann-Roch定理、相交理论；第三章中介绍了曲面的奇点、 纤维化方法和稳定纤维化的周期映射。然后从第四章开始，本书着重讲述曲面的一般性质、 特殊曲面的分类和一般型曲面的典范分类。本书的最后一章中主要介绍曲面的拓扑和微分结构。 笔者认为，本书对于一般型曲面的分类内容略显陈旧。
总的来说，这是一本介绍代数曲面理论的极好工具书。（陈猛）