读GTM52的一些建议

I.什么是GTM52? Graduate texts in mathematics系列，No.52是代数几何学家Hartshorne在1977年发表的一本名为Algebraic geometry的教材。此书一共分为五章：Varieties, Schemes, Cohomology, Curves, Surfaces. 主体部分用Grothendieck的Scheme语言介绍了代数几何中最基本的概念和定理，用上同调的方法给出了一些代 数曲线和代数曲面的经典结论。 II.为什么要读 GTM52经常被称为代数几何的圣经，书中的定义和用法已经成为现代代数几何的规范。书中不到500页的内容里涵盖了极丰富的内容，而且谬误极少，毫无疑问 是数学教材中的经典。代数几何和相关领域的学生，都可以考虑用此书为教材来学习。但在这里必须指出，对于有一定基础的学生，通读此书要一年左右的时间， 所以如果只是想对代数几何有简单了解，有很多更合适的书，我会在最后列出。 III.预备知识在读GTM52之前，需要掌握微分几何，代数拓扑，交换代数的基本知识(比如complex manifolds, vector bundles, (deRham)cohomology, normalization), 尤 其是交换代数，应该掌握Atiyah-Macdonald中的内容。同样对代数几何也应该有简单的了解，GTM52并不适合做学习代数几何的第一本书。推荐给初学者的书 我也会在最后列出。 IV.如何去读因为采用了Grothendieck抽象的Scheme和Category的语言，而且作者很少用文字来解释背后的几何意义，GTM52是非常难上手的一本书。在读的时候切忌从 头到尾，一页一页的读，这是一个老教授让我最受益的忠告（他说”read linearly” is only for undergraduate students）。就是说一定不要读懂了一段再去读下一 段，如果不懂就反复读，读懂了也就不用回头在看了。因为这样读，一本书可能就要读十年了，而且往往也说不清是懂了还是不懂。遇到卡住的时候，只要认真想 过了，尽可以跳过，以后再回头重看。 关于书中的具体内容，第一章是用来热身，其中有很多具体的例子，对后面的理解会很有帮助。读后面时，也应该经常参考这些例子。第二三章最难，肯定要反复 读。这里我建议可以在粗略看过第二章到Differentials, 第三章到Serre duality之后就可以开始读第四章了。因为第二三章建立抽象的理论，在四五章得到应用。 这些应用会帮助读者更具体地理解之前抽象的理论。第二章的Formal schemes在之后的应用不多，可以跳过。第三章Serre duality之后的几节很难，对后来的 内容不是很重要，读第一遍时建议跳过。 每一节后都会有很多习题，应该尽量多做一些（我第一章做了80%，第二三章35%，四五章20%）。作者在定理的证明中经常会引用前面练习题的结果。我建议应 该把这些被引用的习题都做过。做习题时也应该卡住了就跳过，之后再会来重新想。 V.其他参考书目 EGA： 毫无疑问，Grothendieck的EGA是无可替代的经典。虽然GTM52涵盖了EGA大部分的内容，但是Grothendieck超越时代的理解，我们只能从原著中学 习。比如Scheme（X）可以理解成一个从affine schemes(或者rings)到sets的functor, 就是Hom(Spec(R), X). Scheme中的点可以理解成从Hom(Spec(R), X) 中的一个元素，这里的R可以是任意的交换环。这样的解释是以后建立Deformation theory, moduli space的基础。所以这些抽象的Category语言并不仅仅简化 证明，而且让我们从更高的角度来认识问题。但是EGA更加晦涩，可以考虑读过GTM52之后再读。 Basic algebraic geometry，volumn 1,2是Shafarevich在大概同一时期写的一套教材。第一册大概是GTM52第一章的扩充版，第二册现讲Schemes,再讲 Complex manifolds。虽然不深，但是很广，而且论述更详细，是我很喜欢的两本书。和很适合想简单了解代数几何的其他专业的同学读。 Principles of algebraic geometry:是几何大师P. Griffiths和他学生J. Harris写的用复几何来研究代数几何的教材。很漂亮的一本书，而且是这方面唯一的系统 的教材。在A. Weil和O. Zariski之前的数学家就是用这种复几何的语言来研究的，所以读者可以看到早期的代数几何学家的一些想法。在GTM52中，这些想法都 被抽象的语言所掩盖。不过书中小错误很多，不适合刚上手的同学读。有兴趣的同学可以先看看R. O. Wells的Differential analysis on complex manifolds. 另外还有很多不错的代数几何教材，不过我都没怎么看过，这里只列下题目。 David Mumford: complex projective varieties; The red book of varieties and schemes Joe Harris: Algebraic geometry: a first course‎ Joe Harris, David Eisenbud: The geometry of schemes James Milne: Algebraic geometry, lecture notes available at jmilne.org（他有不少笔记，我看过的都很不错） 最后如果有对代数几何感兴趣，但没有很多预备知识（尤其是交换代数）的同学可以考虑看看Riemann surfaces的书。可以感受一下代数几何大概是在研究什么 。

All the way

Just to keep a record of where I am and how I came here.

During the first year at Purdue, I was mainly interested in differential geometry. Since this isn’t very popular at Purdue, I was even thinking about to transfer. My interest in algebra was increasing as I was taking Ulrich’s commutative algebra, and Lipman’s abstract algebra.

Before the first summer, I was looking for some professor to take a book reading course. At first, Dr. Lee told me that she would be gonne for most of the summer. Then I emailed Yeung, Lempert, Donnelly and Catlin, but none of them is available. They are about all faculties around from whom I may learn some differential geometry. So I had to choose something different, and went to Lipman. He agreed to give me a reading course immediately, and later emailed me a book called “Algebraic and analytic geometry” By Aaron Neeman, which was published half year later. The results in the book were very hard. I couldn’t understand the main theorem, but still attracted by the topic. Hence I decided to work on Algebraic geometry.

Yeung is the first person I was thinking to work with. But unfortunately, he already had two students, and didn’t want any more. Under the advice of both Lipman and Yeung, I turned to Arapura, who was on sabatical for the whole year. Still I emailed him and he accepted! During my advisor’s absent, I spent most of the year reading Hartshorne, finished reading the first time around summer 08.

After my advisor came back at fall 09, he wanted me to go over the main part Hartshorne again with him. We spent about 3 months. In November, I took the advanced topic exam with Arapura and Lipman. Two weeks before the exam, Arapura gave a talk on the fundamental groups of compact Kahler manifolds. After that, he asked me some questions, one of them was “can every morphism between finite groups be realized as the induced morphism between fundamental groups from a holomorphic map between compact Kahler manifolds?” I was very interested and was able to solve this problem two days later. So right after the advanced topic exam, I started to work on this subject, with reading Voisin’s paper.

For more than two months, I was studying Kahler manifolds, and it’s topological structures. After the winter break, I switched the topic to vanishing theorems, because there is not a good problem in the previous subject proper for a graduate student. I spend most of the spring semester reading Esnault and Viehweg’s Lectures on vanishing theorems. Right before the Michigan workshop May 09, my advisor told me the specific problem he wanted me to think about. The approach he suggested was such a surprise to me, which also seems to be a long way to go.

Since this summer, I am thinking about either to learn some phisics or some number theory. We’ll see.

Something I want to know

Here I am writing a list of topics I am learning or, most of them, I wish I will have a chance to learn some day.

I should be able to learn (more or less) most of the stuffs(eventually):

1. Vanishing theorem (main job)
2. Positivity in algebraic geometry, Book by Lasarsfeld (working on it)
3. Hodge theory, Deligne’s work (working on it)
4. Minimal model program(the several fundamental theorems of MMP)
5. More transdental method in algebraic geometry
6. Intersection theory
7. Theory of algebraic curves
8. Stacks and moduli spaces
9. Proof of Serre’s GAGA
10. Algebraic K-theory

These are also interesting:

1. Resolution of singularity
2. Grothendieck’s standard conjectures
3. Weil’s conjecture
4. Index theory
5. Chern’s work on sphere bundle
6. Theory of differential operators

current problems

Here are some problems I started to think about recently.

1. Kahler groups: If a group is isomorphic to a fundamental group of a compact Kahler(complex projective) manifold, the we call it “Kahler”(“projective”). In general, we can ask what are the necessary and sufficient conditions to be Kahler or projective. We know projective implies Kahler, since complex projective manifold is always compact Kahler. And we ask, “whether every Kahler group is projective?”
2. Kodaira’s problem: Kodaira proved every compact Kahler surface is equivalent to a projective surface via deformation. And Voisin constructed examples of dim>=4 compact Kahler manifolds, whose cohomology ring will never isomorphic to the one of any projective manifold, hence it is not equivalent to any projective manifold via deformation. But in dimesion 3, the problem is still open.
3. Combining the above two question together, we can also ask the necessary or sufficient conditions of a graded-commutative ring to be Kahler or projective, in the sense of 1.
4. Kahler morphisms: In the sense of 1, what morphisms between groups can be Kahler or projective?

The first three are more difficult and the last one seems more practical. Instead of settle down on one specific problem, I am wandering around and lead by my interest.

Nicolas Bourbaki

Among all the names of 20th century mathematicians, one is very special–Nicolas Bourbaki.

As the author of a series of books on the foundation of modern mathematics, Nicolas Bourbaki was actually a pen name of a group of young, enthusiastic mathematicians in France. They got together about three times a year, and wrote text books, toward the most rigour and generality. And along their way, they also had later very successful seminars.

This group is more or less mysterious, and I do not know many details. However, its spirit, awoke my sleeping passion. Here to present in an accurate way, I will just quote.

“It was soon decided that the work would be collective, without any acknowledgment of individual contributions.”

Borel, Armand (1998). Twenty-Five Years with Nicolas Bourbaki, (1949-1973). Notices Amer. Math. Soc. 45(3): 373-380,

“… …Bourbaki sets off, if you like, from a basic belief, an unprovable metaphysical belief we willingly admit. It is that mathematics is fundamentally simple and that for each mathematical question, there is, among all the possible ways of dealing with it, a best way, an optimal way.”

J. DIEUDONNé: The Work of Nicholas Bourbaki, American Math. Monthly 77,1970, pp134-145

In math, there are always people, idealistic, pursuing the best and the optimal. Even being unsuccessful, they dream, they walk, they fight, they laugh…

Encounter complex

Recently, I saw some wonderful things about complex structures.

The reason I start thinking about complex manifolds seriously was the interest arisen from my adviser’s seminar about 3 weeks ago. Where he asked the question to understand the possible groups which can be realized as fundamental groups of compact Kahler manifolds or projective smooth varieties over complex numbers. Investigation of properties distinguishing compact Kahler manifolds and projective varieties leaded me to some recent results of Claire Voisin.

In 1950s, Kodaira proved that any compact Kahler surface is deformation equivalent to a complex projective surfaces, by classification of compact Kahler surfaces. And people began to think about higher dimensional cases, whether all compact Kahler manifolds are equivalent to some projective ones, which is called Kodaira’s problem. Voisin, in her 2005 paper, gave counter examples for dimension>=4 cases. In dimension 3, problem is still open.

The essence Voisin created her examples is through blowing up some lower dimesional subspaces of a product of tori to encode information in the first cohomology group, from which to recover an endomorphism of original torus. A direct generalization to 3-dimesional case seems impossible, because of the lack of submanifolds of a low dimesional non-projective torus.

Alexander Grothendieck

Alexander Grothendieck，1928年生于德国柏林，1939年移居法国，在(Laurent Schwartz)的指导下于1953年完成博士学位，成为拓扑向量空间的专家。但1957年后，他转向代数几何和同调代数的研究。因为在代数几何，同调代数和泛函分析中的杰出贡献，他在1966年被授予费尔斯奖。

他在代数几何上的贡献是革命性的。他在法国的讨论班几乎统治了60年代的代数几何界，以这个讨论班为基础出版的EGA(Éléments de géométrie algébrique)和SGA(Séminaire de géométrie algébrique)构建了近代代数几何的基础。他首先提出的scheme理论已经成为代数几何的基本工具；他在同调论上的结论构建了连接代数几何与数论的桥梁。他受邀请在1958年的国际数学家大会上做一小时关于代数几何与同调论的报告（四年一届的国际数学家大会每次大概会邀请十位数学家做一小时的报告，这被认为是特别的荣誉），在报告中他提出了著名的Grothendieck Duality，但由于当时数学工具，或者说数学语言的限制，他无法把结论严格的表示出来。之后经过两年时间，他从建立需要的基础开始，完成的Grothendieck Duality的完整陈述和证明。

所有这些成就，全部在他四十二岁前完成。1970年，由于发现学校IHES(Institut des Hautes Études Scientifiques)的资金部分来自军方，他决定退出科学界。辞职后，他开始宣传他的反战与生态保护思想。在1988年他拒绝了瑞典皇家科学院授予的Crafoord奖。1991年，他离开他的家，并从此消失。据传说，他之后定居在法国南部或安道尔过着与世隔绝的生活。

Purdue的Professor Lipman对他的评价是：他思考的速度是正常人的十倍。可能也是因为他思考得太快，太多，后来好像有些疯狂了。

他的作风一般人的确很难理解，后来他自己也在拒绝别人的建议发表文集的时候说过，他的思想没有真正被世人理解。

他认为研究数学唯一的动力是对未知的探索，而因为过多名利的卷入使数学界变得不纯洁。现在全世界可能都找不到任何能达到这种境界的学者了，但是我非常赞成他的观点，以为具有永恒价值的，永远不是名和利。

tame topology：一般定义的拓扑在某些时候会有一些不理想的性质，例如通过连续映射可以用直线覆盖平面。因此Grothendieck提出了tame topology的概念，从新定义了一种新的拓扑概念。在更强的限制下，新的拓扑结构会有更好的性质，从而成为被代数几何学者经常使用的概念。

Galois, Évariste (1811-1832)

在数学的世界里，从来不缺少天才和奇迹。我们的故事就是关于这样一个创造奇迹的天才。

伽罗华，埃瓦利斯特(Galois, Évariste)出生在1811年的法国，拿破仑统治下充满战争和政治动荡的国家。他童年的生活非常平淡，在十一岁之前，她的母亲是他唯一的老师。十二岁之后他进入了巴黎的正式学校，并以优秀的成绩完成了前两年的学习。不过很快他就厌倦了正常的功课，而开始自己钻研拉格朗日(Joseph Louis Lagrange)和阿贝尔(Niels Henrik Abel)在代数上的工作。他的物理老师曾经对他作出过这样的评价：“他几乎什么都不知道。有人告诉我说她很厚数学才华，这让我很惊讶。根据他的考试成绩，我不认为他有任何才智，或者是他把才智隐藏得太好了，以至于我完全察觉不到。”在此之前他已经发表了两篇论文，并且开始了对一元多项式求解的探索。在两次申请被巴黎综合理工学院拒绝之后，他进入了高等师范学校。

在十七岁时，他将自己的两篇论文寄给法国科学院要求发表，但作为当时全世界最著名的数学家柯西(Augustin Louis Cauchy)因为叙述不清晰的原因拒绝发表。之后十九岁的伽罗华加入了国家防卫队，因为过激的行为两次入狱。但这些都没能影响到他继续发展数学上已有的成果。

之前无论发生过什么，当他决定为自己心爱的女子参加和一位士兵的决斗时已经不重要了。自知必死的伽罗华在决斗前夜拼命地把自己的全部成果都狂书下来，并不时地在一边写下“我没有时间”。他在决斗中中枪，第二天，二十岁的伽罗华在医院中去世。他留给哥哥Alfred的最后一句话是：“不要哭泣，Alfred！我要自己全部的勇气在二十岁便死去。”

伽罗华算不上贡献最大的数学家，但他的思想是超越时代的。当时最伟大的两位数学家柯西和泊松(Simeon Denis Poisson)都认为他的论文是不可理解的。直到他去世十一年之后，刘维尔(Joseph Liouville)重新阅读了他的手稿，承认结果的正确性后，将其发表。之后人们开始发现他决斗前夜写下的三十二页手稿意味深长，颇具价值。这些手稿构建了Galois Thoery的基础，甚至可以说是近世代数的基础。理解伽罗华的思想，用去了后人几十年的时间。

在我心目中，另一位超越时代的数学家是近代代数几何的奠基人之一法国人Grothendick。

Reference: wikipedia, mathworld, 数学传播