小平邦彦Kunihiko Kodaira

小平邦彦的父母是权一（1884-1976）和一小平邦彦（1894-1993）。小平邦彦的父亲权一曾在东京帝国大学学习农业和政治学，在他儿子出生时，他正在农业部工作。他于1939年从农业部退休，并当选为日本国会议员，在第二次世界大战期间任职。日本战败后，盟军解除了他的公职。除了这些活动之外，他还撰写了约40部学术著作和350篇学术论文。小平邦彦的母亲一，是校长金井久治的女儿。小平邦彦是他父母所生儿子中的长子，有一个弟弟信彦（生于1919年）。

小平邦彦于1921年进入小学，但对这个小男孩来说，这些年并不轻松。他相当害羞，说话时口吃，尤其是在压力之下。他肯定不是运动型的人，因此，他非常讨厌体育课。在他的自传[6]中，他说自己在小学时是个差生，尽管总的来说他过于谦虚，但他在这个阶段可能确实不出色。然而，他从小就对数字着迷，喜欢数豆子，十岁时他做了一个实验，看看他的狗会不会数数。当狗生了小狗时，小平邦彦把小狗藏了起来，狗很不安地寻找，直到他把小狗还给她。然而，当他藏起几只小狗时，狗似乎很高兴，所以十岁的小平邦彦断定“狗不会数数”。小平邦彦的父亲权一在1921-22年间在德国待过，由于德国通货膨胀失控，他发现坚挺的日元可以非常便宜地购买大量商品。他给孩子们带回了许多礼物，年幼的小平邦彦从父亲给他的精美德国拼装玩具中获得了许多乐趣。这个玩具让小平邦彦在年幼时就决定要成为一名工程师。

小平邦彦于1927年完成小学教育，进入中学。他在[6]中声称自己是个差劲的中学学生，但这与事实不太相符。他在英语课和数学上表现良好，很快就远远超过了同学。当他完成三年课程的一半时，他已经学完了算术、代数、二维和三维几何的全部教学大纲，并解决了指定教材中的所有问题。因此，他购买了藤原松三郎的Algebra，这是一本大学本科教材，并开始钻研矩阵、行列式、连分数、二次互反律和其他概念。我们注意到藤原松三郎（1881-1946）是一位博学多产的作者，于1928-29年出版了两卷本专著Daisugaku（代数）。有人将这本高质量的教材与约瑟夫·阿尔弗雷德·塞雷和海因里希·马丁·韦伯的经典著作相提并论，值得一提的是，他曾在巴黎、哥廷根和柏林学习过。

小平邦彦的父亲在1921-22年德国之行中带回的许多东西之一是一架钢琴。十五岁时，小平邦彦开始学习弹钢琴，并请东京大学的学生中岛先生担任他的钢琴老师。当中岛先生大学毕业搬走后，他的妹妹中岛田津子担任了小平邦彦的音乐老师，尽管她是一名小提琴手而非钢琴家。中学毕业后，小平邦彦进入第一高等学校学习，师从荒又秀夫（1905-1947），他是一位优秀的数学家，写过关于矩阵和行列式的书，以及关于zeta函数的有趣论文。小平邦彦看到荒又多么享受数学，意识到这就是适合他的学科。他在这个教育阶段决定要成为一名数学教师。

1935年，小平邦彦在东京大学开始他的大学教育。在学习的头一年，他参加了高木贞治讲授的“分析导论”课程。这是高木贞治在1936年退休前最后一年授课。弥永昌吉，[13]的作者，为这门课程主持了习题课。他写道[13]：-

我曾给出一个习题问题，证明自然指数函数的底e不是二次无理数（在讲座中已证明e是无理数之后）。小平邦彦走到黑板前，一言不发地用几行写下了他的证明。与其他学生一起阅读这些行时，我们钦佩他完美的证明，其中每个字都切中要害！

末纲恕一于1936年高木贞治退休时被任命为东京大学算术和代数讲席。1936-37年，小平邦彦参加了弥永昌吉基于冯·诺伊曼思想的现代分析课程。他还参加了末纲恕一的讲座，并在1937年底联系末纲恕一，请求在下一年被接纳进入他的讨论班。他被接受了，但后来末纲恕一写信给他，建议他认为在弥永昌吉的讨论班学习几何会更合适。他加入了弥永昌吉的讨论班，并在1937-38年经常在弥永昌吉家中。在那里他弹钢琴，展示了他作为钢琴家的巨大天赋。弥永昌吉的妹妹Seiko也是一位热衷音乐的人，是一名小提琴学生，师从Tazuku Nakajima。小平邦彦于1938年3月从东京大学毕业，获得数学学士学位。不满足于一个学位，他于1941年3月从东京大学物理系毕业，获得物理学学士学位。我们注意到，到1941年他已经发表了十篇论文。在这些年学习物理期间，他与弥永昌吉家族变得更加友好。除了弥永昌吉的女儿Seiko外，家里还有两个男孩。两人一生都取得了很大成就：Kyojiro 弥永昌吉成为尼康光学公司总裁，Teizo 弥永昌吉成为东京大学日本史教授。Tazuku Nakajima组织音乐会，小平邦彦为小提琴手钢琴伴奏。他为在这些音乐会上演奏的Seiko伴奏，两人变得亲密。他们于1943年结婚，去Gora度蜜月。Gora是箱根附近的一个温泉度假胜地，位于本州中南部，坐落在芦之湖南岸，在死火山箱根山破火山口内。他们的蜜月并不像听起来那么田园诗般，因为日本处于战争状态，食物非常稀缺，这对夫妇不得不自带大米到他们住的酒店，因为酒店的厨房空空如也。1944年3月，他们的第一个孩子，一个他们命名为Kazuhiko的男孩出生了，但随着日本遭受严重攻击，东京的条件变得越来越困难。不幸的是，Kazuhiko患上了肾脏问题，于1946年去世。Kodaira夫妇还有两个女儿，Yasuko和Mariko。

小平邦彦于1941年4月被任命为东京帝国大学物理系的讲师，随后于1942年4月被任命为东京文理大学数学系的副教授。除了后一个职位外，他于1944年4月被提升为东京帝国大学物理系的副教授。到1944年秋天，东京被认为对家庭来说太危险，所有的妇女和儿童都搬到了东京以北遥远山区的轻井泽镇的安全地带。在小平邦彦于秋季学期末完成在东京的教学后，他与家人团聚在轻井泽。随着东京在1945年2月遭到超过1000架美国飞机的猛烈轰炸，物理和数学研究所被疏散。4月13日，一次空袭摧毁了他们在东京的家，小平邦彦和他的家人搬到了他父亲拥有房子的米泽。1945年8月6日，一颗原子弹被投在广岛，8月9日第二颗原子弹被投在长崎。日本于8月14日向盟军投降，那年秋天晚些时候，物理和数学研究所在东京重新开放。小平邦彦在研究所重新开放几周后返回，将家人留在米泽。相当令人惊讶的是，小平邦彦能够迅速重新开始他的讨论班，并再次开始产生显著的结果。然而，他在[6]中写道：-

我曾以为会永远生活在日本，享受数学和音乐。这个想法被战争彻底摧毁了。

此时小平邦彦对拓扑学、Hilbert spaces、阿尔弗雷德·哈尔测度、李群和几乎周期函数感兴趣。当然，第二次世界大战对日本产生了严重影响，特别是它有效地将日本科学家与世界其他科学家的联系隔离开来。尽管如此，小平邦彦还是能够获得阅读数学发展的论文，他受阅读赫尔曼·外尔、Stone、冯·诺伊曼、威廉·瓦兰斯·道格拉斯·霍奇、安德烈·韦伊和奥斯卡·扎里斯基的著作影响最大。小平邦彦于1949年4月因他的学位论文Harmonic Fields in Riemannian Manifolds获得东京大学的博士学位，并于1949年在Annals of Mathematics上发表了一篇80页的论文。通过这篇论文，他为世界各地的数学家所熟知，特别是凭借这篇论文，他收到了赫尔曼·外尔的邀请来到普林斯顿。唐纳德·斯宾塞写道[21]：-

这篇论文也打动了其他人，包括我，我邀请小平邦彦在1949-1950学年在普林斯顿大学就他的论文进行演讲。这是合作的开始，产生了十二篇论文，以及我们直到他最近去世的亲密友谊。

小平邦彦接受了赫尔曼·外尔的邀请，从1949年9月起，他在普林斯顿高等爱德华·斯图迪研究院担任研究员一年。此后，他于1950年9月至1951年6月在约翰斯·威廉·霍普金斯大学担任访问教授，之后回到普林斯顿高等斯图迪研究院。此时，他的妻子清子以及两个年幼的女儿靖子和真理子（此前一直留在日本）来到普林斯顿与他团聚。他于1952年9月被任命为普林斯顿大学副教授，并于1955年9月晋升为正教授。在此之前，他一直保留着在东京的职位，但在被任命为普林斯顿正教授后，他辞去了东京的职务。迈克尔·阿蒂亚写道，小平邦彦在这一时期做出了卓越的数学工作[8]：-

在普林斯顿期间，小平邦彦继续参与调和形式的研究，特别是在它们对代数几何的应用中，这一领域也为威廉·瓦兰斯·道格拉斯·霍奇的工作提供了动力。20世纪50年代见证了复代数几何的繁荣，其中层论的新方法，起源于法国的让·勒雷、埃利·嘉当和让-皮埃尔·塞尔之手，提供了一整套处理全局问题的新机制。层论与威廉·瓦兰斯·道格拉斯·霍奇理论相契合，因此小平邦彦自然处于利用新发展的有利位置。他做到了这一点，在一系列与唐纳德·斯宾塞合作撰写的论文中迅速接连发表。这些论文改变了代数几何的面貌，并提供了框架，使弗里德里希·希策布鲁赫和其他年轻一代能够取得惊人的进展。大量被古典学派的意大利几何学家留下未解决或不完整的问题，现在以令人信服的方式得到了处理。

这项工作使小平邦彦在1954年被提名为菲尔兹奖。他于八月中旬从纽约启航，前往参加1954年9月在阿姆斯特丹举行的国际数学家大会。在9月2日的开幕式上，赫尔曼·外尔向他颁发了约翰·查尔斯·菲尔兹奖章，另一位菲尔兹奖得主让-皮埃尔·塞尔也同样获此殊荣。小平邦彦于9月3日向大会作了题为“代数簇超越理论中的一些结果”的报告。然而，当他回到美国时，他发现普林斯顿的条件并不完全合他心意。他写道[6]：-

由于所罗门·莱夫谢茨已经退休，我逐渐意识到普林斯顿的资深教授们讨厌我。

在应奥斯卡·扎里斯基之邀于1961年9月起在哈佛大学担任访问教授一年后，他于1962年9月被任命为约翰斯·霍普金斯大学数学讲席教授。1965年，小平邦彦离开约翰斯·霍普金斯大学，出任斯坦福大学数学讲席教授。唐纳德·斯宾塞对普林斯顿没有试图挽留小平邦彦留在该校任教感到非常愤怒，以至于他从普林斯顿辞职并搬到斯坦福，以便与小平邦彦在一起。在斯坦福期间，小平邦彦讲授了抽象复解析流形研究的导论，他的课程被整理成书Complex manifolds（1971年）。在斯坦福两年后，他回到日本，自1967年起担任东京大学数学讲席教授[8]：-

小平邦彦回到日本后，他讲课并主持讨论班，吸引了许多有才华的学生。小平邦彦的影响如此显著，可以说他建立了一个日本代数几何学家的新学派。

在东京大学，他于1971年至1973年担任理学部部长，之后于1975年3月退休。我们应当注意到，他并不情愿担任部长，因为数学系曾向他保证，当他在1967年回到东京大学时，不会给他任何行政职务。理学部与小平邦彦没有这样的协议，并在他极不情愿的情况下选举他为部长。他是一位出色的部长，但讨厌这个角色。他担任部长期间的一个后果是他停止了研究工作。即使在两年后辞去部长职务之后，他也从未重新开始研究。这解释了他的自传性著作[7]Notes on an idle mathematician的标题。

小平邦彦的工作涵盖许多主题。这些包括将大卫·希尔伯特空间方法应用于微分方程，这在他早期工作中是一个重要主题，并且很大程度上是受赫尔曼·外尔影响的结果。这一次，在威廉·瓦兰斯·道格拉斯·霍奇的影响下，他研究调和积分，后来他将这项工作应用于代数几何中的问题。小平邦彦工作的另一个重要领域是将层应用于代数几何。大约在1960年，他开始参与紧复解析空间的分类。贯穿他大部分工作的一个主题是波恩哈德·黎曼-Roch定理，这在他的许多研究中起着重要作用。

小平邦彦因其杰出的研究获得了许多荣誉。也许最值得一提的是1954年获得的约翰·查尔斯·菲尔兹奖章，我们已经提到过，但他还在1957年获得了日本科学院的日本学士院奖，并于同年获得日本政府颁发的文化勋章。他于1975年获得著名的藤原奖，并于1984年获得沃尔夫基金会数学奖。沃尔夫奖的颁奖词指出，该奖授予小平邦彦：-

……因他在复流形与代数簇研究方面的杰出贡献。小平邦彦教授对调和积分作了深刻研究，并卓有成效地应用于代数几何与复几何，其中包括射影嵌入定理、复结构的形变（与唐纳德·斯宾塞合作）以及复解析曲面的分类。他的工作极大地影响和启发了全世界这些领域的研究者。

他被世界各地多个科学院和学术团体聘为荣誉成员，其中包括哥廷根科学院（1974年）、National Academy of Sciences（1975年）、American Academy of Arts and Sciences（1978年）和伦敦数学会（1979年）。

1975年从东京大学退休后，他被聘为学习院大学理学部教授，这是一所评价很高的私立大学。他在该校任教十年，但越来越担忧学生水平的下降。这促使他写信给文部省[14]：-

……指责文部省压制个性，扼杀儿童和大学生的创造力与主动性……

并编写中小学和大学教科书，试图提高数学教学水平。例如，1977年他写了Complex analysis （日文版），该书被译成英文并于2007年出版。出版商写道：-

这本教科书由该学科的大师撰写，将受到学生和专家的赞赏。作者以清晰直接的方式展开经典的复变量函数理论。总体而言，这里采用的方法强调该理论的几何方面，以避免与该主题相关的一些拓扑陷阱。因此，奥古斯丁·路易·柯西积分公式首先在一个拓扑上简单的情形中得到证明，作者由此推导出全纯函数的基本性质。从基础出发，引导学生学习共形映射、波恩哈德·黎曼映射定理、波恩哈德·黎曼曲面上的解析函数，最终学习波恩哈德·黎曼-Roch定理和尼尔斯·阿贝尔定理。本书插图丰富，包含大量例题和习题（其中许多附有解答），应能成为复分析高级课程的一本富有启发性的教材。

1979年，他用日语出版了五卷本的Introduction to analysis，内容涵盖实数、函数、微分、积分、无穷级数、多元函数、曲线与曲面、Fourier series、约瑟夫·傅里叶变换、常微分方程以及广义函数。1986年，他出版了专著Complex manifolds and deformation of complex structures。Andrew Sommese 以如下文字开始了一篇详细的评论：-

在数学和科学中，对象（例如方程组）依赖于参数是常见现象。对这种依赖性的研究因领域不同而有多种名称，如分岔研究、展开研究或形变研究。从历史和概念上看，紧复流形的局部形变理论在现代对这些现象的理解中扮演了核心角色。《复流形与复结构的形变》是该理论创始人之一对其局部紧复解析形变理论的精心阐述。

James Carlson 在评论同一本书时写道：-

作者与唐纳德·斯宾塞共同创立了复流形形变理论，他撰写的这本书将为所有对这一如今已十分庞大的主题感兴趣的人提供帮助。

他生命中的最后十年是在与健康问题作斗争中度过的。他患有呼吸系统疾病，并且变得非常耳聋，这让他极为伤心，因为他无法再欣赏音乐，而音乐在他一生中对他意义重大。1990年，他因病情太重而未能出席在京都举行的国际数学家大会。弗里德里希·希策布鲁赫在[12]中回忆了他与小平邦彦的最后几次会面：-

小平邦彦是我的朋友和老师。我和妻子记得我们最后一次拜访小平家在东京的住所。他正在厨房的桌子上编写中学教科书。节子小平邦彦准备饭菜时不得不把那些纸张推开。1995年，我祝贺他八十岁生日。他以他迷人的方式回了信。但当我们1996年来到东京时，他已经住院了。我们再也无法与他交谈了。

小平邦彦的妻子Seiko于2000年1月去世，距她丈夫去世两年半。

Kunihiko Kodaira's parents were Gon-ichi (1884-1976) and Ichi Kodaira (1894-1993). Gon-ichi, Kunihiko's father, had studied agriculture and politics at Tokyo Imperial University and, at the time his son was born, was working at the Ministry of Agriculture. He retired from the Ministry of Agriculture in 1939 and was elected to the Japanese Parliament where he served during World War II. After Japan was defeated, the Allies removed him from public office. In addition to these activities, he wrote around 40 academic books and 350 academic papers. Ichi, Kunihiko's mother, was the daughter of the schoolmaster Kyuji Kanai. Kunihiko was the eldest of his parents' sons, having a younger brother Nobuhiko (born 1919).

Kunihiko entered elementary school in 1921 but these were not easy years for the young boy. He was rather shy and stammered when he spoke, especially when he was under stress. He was certainly not the athletic type and, as a consequence, he had a strong dislike of the physical education classes. In his autobiography [6] he says that he was a poor pupil in primary school and, although on the whole he is overly modest, nevertheless he probably did not shine at this stage. However, he showed a fascination with numbers from a young age, loved counting beans and, when he was ten years old he conducted an experiment to see if his dog could count. When she produced puppies, Kunihiko hid them and the dog was upset searching until he returned them to her. However, when he hid a couple of the puppies the dog seemed happy so the ten year old Kunihiko decided that "dogs can't count". Gon-ichi, Kunihiko's father, spent time in Germany in 1921-22 and, with German inflation out of control, he found that the strong Japanese yen could buy large quantities of goods very cheaply. He brought back to Japan many gifts for his children and the young Kunihiko had much enjoyment from the wonderful German construction kits his father gave him. This toy made Kunihiko decide at a young age that he wanted to be an engineer.

Kodaira completed his primary education in 1927 and entered the middle school. He claims in [6] that he was a poor middle school pupil but this does not quite fit the facts. He did well in English classes and in mathematics, soon getting far ahead of his fellow pupils. By the time he had completed half of the three year course, he had covered the whole syllabus of arithmetic, algebra, 2- dimensional and 3-dimensional geometry and solved all the problems in the set text. He therefore purchased Algebra by M Fujiwara, an undergraduate university text, and began to work his way through matrices, determinants, continued fractions, quadratic reciprocity and other concepts. We note that Fujiwara Matsusaburo (1881-1946) was an erudite and prolific author who published the two-volume treatise Daisugaku (Algebra) in 1928-29. Some have compared this quality text with the classic books by Joseph Serret and Heinrich Weber and it is worth mentioning that he had studied in Paris, Göttingen and Berlin.

One of the many things that Kodaira's father had brought back from his 1921-22 trip to Germany was a piano. When he was fifteen years old, Kodaira began to learn to play the piano and had a student from Tokyo University, Mr Nakajima, as his piano teacher. When Mr Nakajima graduated from the university and moved away, his sister Tazuku Nakajima acted as Kodaira's music teacher although she was a violinist rather than a pianist. After middle school, Kodaira studied at the First High School where he was taught by Hideo Aramata (1905-1947) who was an excellent mathematician writing books on matrices and determinants as well as interesting papers on the zeta-function. Kodaira saw how much Aramata enjoyed mathematics and realised that it was the subject for him. He decided at this stage of his education that he wanted to become a mathematics school teacher.

In 1935 Kodaira began his university education at the University of Tokyo. In his first year of study he took the course 'Introduction to Analysis' given by Teiji Takagi. This was Takagi's last year lecturing before he retired in 1936. Shokichi Iyanaga, the author of [13], conducted the exercise class for this course. He writes [13]:-

I had given as an exercise problem to prove that the base e of the natural exponential function is not an irrational of the second degree (after it had been proved in a lecture that e is irrational). Kodaira came to the blackboard and wrote his proof in a few lines without speaking any word. In reading these lines with the other students, we admired his perfect proof, where every word was to the point!

Zyoiti Suetuna was appointed to the chair of arithmetic and algebra at the University of Tokyo in 1936 when Takagi retired. In 1936-37 Kodaira attended Iyanaga's course on modern analysis which was based on the ideas of von Neumann. He also attended lectures by Suetuna and, towards the end of 1937, approached Suetuna asking to be admitted to his seminar in the following year. He was accepted but later Suetuna wrote to him suggesting that he thought that studying geometry in Iyanaga's seminar would be more appropriate. He joined Iyanaga's seminar and, in the year 1937-38, he was often in Iyanaga's home. There he played the piano showing his great talents as a pianist. Iyanaga's sister Seiko was also a keen musician, and was a student of the violin studying with Tazuku Nakajima. Kodaira graduated from the University of Tokyo in March 1938 with a Bachelor of Science in mathematics. Not content with one degree, he graduated from the physics department at the University of Tokyo in March 1941 with a Bachelor of Science in physics. We note that by 1941 he already had ten papers published. During these years while he studied physics he became more friendly with the Iyanaga family. In addition to Seiko, Shokichi Iyanaga's daughter, there were two boys in the family. Both achieved much in their lives: Kyojiro Iyanaga became president of Nikon Optics, and Teizo Iyanaga became a professor of Japanese history at the University of Tokyo. Tazuku Nakajima organised concerts and Kodaira accompanied the violinists on the piano. He accompanied Seiko, who played in these concerts, and the two became close. They married in 1943 and went to Gora for their honeymoon. Gora is a hot spring resort near Hakone, in south-central Honshu, lying on the south bank of Lake Ashino, in the caldera of the extinct volcano Mount Hakone. Their honeymoon was not quite so idyllic as this sounds since, with Japan at war, food was very scarce and the couple had to bring their own rice to the hotel they stayed in as its kitchens were bare. In March 1944 their first child, a boy they named Kazuhiko, was born but conditions in Tokyo became steadily more difficult as Japan came under severe attacks. Sadly, Kazuhiko developed kidney problems and died in 1946. The Kodairas also had two daughters, Yasuko and Mariko.

Kodaira had been appointed as a Lecturer in the Department of Physics of the Imperial University of Tokyo in April 1941 and then as an Associate professor in the Department of Mathematics of Tokyo Bunri University in April 1942. In addition to this latter post, he was promoted to Associate professor in the Department of Physics of the Imperial University of Tokyo in April 1944. By the autumn of 1944 Tokyo was considered too dangerous for the family and all the women and children moved to the safety of the town of Karuizawa, in the mountains far north of Tokyo. After Kodaira completed his teaching in Tokyo at the end of the autumn term, he joined his family in Karuizawa. As Tokyo came under heavy attack with over 1000 US planes bombing the city in February 1945, the Physics and Mathematics Institutes were evacuated. On 13 April, an air raid destroyed their home in Tokyo and Kodaira and his family moved to Yonezawa where his father owned a house. On 6 August 1945 an atomic bomb was dropped on Hiroshima and on 9 August a second atomic bomb was dropped in Nagasaki. Japan surrendered to the Allies on 14 August and, later that autumn the Physics and Mathematics Institutes reopened in Tokyo. Kodaira returned to the Institutes a couple of weeks after they reopened, leaving his family in Yonezawa. Rather amazingly, Kodaira was able to quickly restart his seminar and began again producing remarkable results. However, he writes in [6]:-

I had thought to live always in Japan, enjoying mathematics and music. This thought was completely destroyed by the War.

At this time Kodaira was interested in topology, Hilbert spaces, Haar measure, Lie groups and almost periodic functions. Of course, World War II had a severe affect on Japan, in particular it effectively isolated Japanese scientists from contacts with other scientists around the world. Despite this Kodaira was able to obtain papers to read of mathematical developments and he was most influenced by reading the works of Weyl, Stone, von Neumann, Hodge, Weil and Zariski. Kodaira was awarded his doctorate from the University of Tokyo in April 1949 for his thesis Harmonic Fields in Riemannian Manifolds and published it in an 80-page paper in the Annals of Mathematics in 1949. Through this paper he became well known to mathematicians world-wide and, in particular, on the strength of this paper he received an invitation from Weyl to come to Princeton. Donald Spencer writes [21]:-

This paper also impressed others, including me, and I invited Kodaira to lecture on his paper at Princeton University during the academic year 1949-1950. This was the beginning of a collaboration which resulted in twelve papers and our close friendship extending to his recent death.

Kodaira accepted Weyl's invitation and, from September 1949, he spent a year as a fellow of the Institute for Advanced Study at Princeton. Following this he was a Visiting professor at Johns Hopkins University from September 1950 to June 1951 when he returned to the Institute for Advanced Study at Princeton. At this time his wife Seiko and their two young daughters Yasuko and Mariko, who had remained in Japan untill then, joined him in Princeton. He was appointed as an Associate professor at Princeton University in September 1952 and was promoted to a full professorship there in September 1955. Up to this time he had kept open his position in Tokyo but, after being made a full professor at Princeton he resigned his positions in Tokyo. Michael Atiyah writes about the remarkable mathematics that Kodaira produced in this period [8]:-

During his time at Princeton, Kodaira continued his involvement with harmonic forms, particularly in their application to algebraic geometry, the area which had also provided the motivation for Hodge's work. The 1950s saw a great flowering of complex algebraic geometry, in which the new methods of sheaf theory, originating in France in the hands of Leray, Cartan and Serre, provided a whole new machinery with which to tackle global problems. Sheaf theory fitted with Hodge theory, so it was natural that Kodaira should have been well placed to exploit the new developments. This he did, in a rapid succession of papers written in collaboration with Donald Spencer. These papers altered the face of algebraic geometry, and provided the framework in which Hirzebruch and others of the younger generation were able to make spectacular progress. Large numbers of problems left unsolved or incomplete by the Italian geometers of the classical school were now disposed of in convincing fashion.

This work led to Kodaira being nominated for a Fields Medal in 1954. He sailed from New York in mid August to travel to the International Congress of Mathematicians to be held in Amsterdam in September 1954. He was presented with the Fields Medal by Hermann Weyl at the opening ceremony on 2 September, as was the other Field Medalist Jean-Pierre Serre. Kodaira delivered his lecture 'Some results in the transcendental theory of algebraic varieties' to the Congress on 3 September. However, when he returned to the United States he did not find conditions at Princeton entirely to his liking. He wrote [6]:-

Since Lefschetz had retired, I gradually realised that the older professors at Princeton hated me.

After spending a year as a visiting professor at Harvard from September 1961 following Oscar Zariski's invitation, in September 1962 he was appointed to the chair of mathematics at Johns Hopkins University. In 1965 Kodaira left Johns Hopkins to take up the chair of mathematics at Stanford University. Donald Spencer was so angry that Princeton had not made an attempt to keep Kodaira on the faculty there, that he resigned from Princeton and moved to Stanford to be with Kodaira. While at Stanford Kodaira gave an introduction to the study of abstract complex analytic manifolds and his course was written up as the book Complex manifolds (1971). After two years at Stanford, he returned to Japan and held the chair of mathematics at the University of Tokyo from 1967 [8]:-

After Kodaira's return to Japan, he gave lectures and ran seminars which attracted many able students. Kodaira's influence was so pronounced that one could say that he established a new school of Japanese algebraic geometers.

At the University of Tokyo he served as Dean of the Faculty of Science 1971-73 before retiring in March 1975. We should note that he was a reluctant Dean for he had been given a guarantee by the Mathematics Department that he would not be given any administrative duties when he returned in to the University of Tokyo in 1967. The Faculty of Science had no such agreement with Kodaira and elected him Dean much against his wishes. He was an excellent Dean but hated the role. A consequence of his period as Dean was that he stopped doing research. He never restarted even after he resigned as Dean after two years. This explains the title of his autobiographical work [7] Notes on an idle mathematician.

Kodaira's work covers many topics. These include applications of Hilbert space methods to differential equations which was an important topic in his early work and was largely the result of influence by Weyl. This time through the influence of Hodge, he worked on harmonic integrals and later he applied this work to problem in algebraic geometry. Another important area of Kodaira's work was to apply sheaves to algebraic geometry. In around 1960 he became involved in the classification of compact, complex analytic spaces. One of the themes running through much of his work is the Riemann-Roch theorem and this plays an important role in much of his research.

Kodaira received many honours for his outstanding research. Perhaps the most noteworthy was the award of a Fields Medal in 1954 which we have already mentioned but he also received the Japan Academy Prize from the Academy of Japan in 1957 and the Order of Culture from the Japanese Government in the same year. He received the prestigious Fujiwara Prize in 1975 and the Wolf Foundation Prize in Mathematics in 1984. The citation for the Wolf Prize states that the prize was awarded to Kunihiko Kodaira:-

... for his outstanding contributions to the study of complex manifolds and algebraic varieties. ... Professor Kunihikio Kodaira made a profound study of harmonic integrals with incisive, important applications to algebraic and complex geometry. These include the projective imbedding theorem, deformations of complex structures (with D C Spencer), and the classification of complex analytic surfaces. His work has greatly influenced and inspired researchers in these subjects throughout the world.

He was made an honorary member of several academies and learned societies throughout the world, including the Göttingen Academy of Sciences (1974), the National Academy of Sciences (1975), the American Academy of Arts and Sciences (1978) and the London Mathematical Society (1979).

After retiring from the University of Tokyo in 1975, he was appointed as Professor in the Faculty of Science of Gakushuin University, a highly-rated private university. He taught for ten years at this university but became increasingly worried about falling standards of the students. This led him to write to the Ministry of Education [14]:-

... accusing the Ministry of Education of crushing individualism, and eliminating creativity and initiative in children and university students ...

and to write school and university textbooks to try to improve the standard of mathematics teaching. For example, in 1977 he wrote Complex analysis (Japanese) which was translated into English and published in 2007. The publisher writes:-

Written by a master of the subject, this textbook will be appreciated by students and experts. The author develops the classical theory of functions of a complex variable in a clear and straightforward manner. In general, the approach taken here emphasises geometrical aspects of the theory in order to avoid some of the topological pitfalls associated with this subject. Thus, Cauchy's integral formula is first proved in a topologically simple case from which the author deduces the basic properties of holomorphic functions. Starting from the basics, students are led to the study of conformal mappings, Riemann's mapping theorem, analytic functions on a Riemann surface, and ultimately the Riemann-Roch and Abel theorems. Profusely illustrated and with plenty of examples, and problems (solutions to many of which are included), this book should be a stimulating text for advanced courses in complex analysis.

In 1979 he published the five volume Introduction to analysis in Japanese covering real numbers, functions, differentiation, integration, infinite series, functions of several variables, curves and surfaces, Fourier series, Fourier transforms, ordinary differential equations, and distributions. In 1986 he published the monograph Complex manifolds and deformation of complex structures. Andrew Sommese begins a detailed review as follows:-

In mathematics and science it is a familiar occurrence to have objects, such as systems of equations, depending on parameters. The investigation of this dependence goes under many names such as the study of bifurcations, or of unfoldings, or of deformations depending on the area. Historically and conceptually, the local deformation theory of compact complex manifolds has played a central role in the modern understanding of these phenomena. 'Complex manifolds and deformation of complex structures' is a careful exposition of this local compact complex analytic deformation theory by one of its founders.

James Carlson, reviewing the same book, writes:-

The author, who with Spencer created the theory of deformations of a complex manifold, has written a book which will be of service to all who are interested in this by now vast subject.

The last ten years of his life were ones during which he battled against health problems. He suffered from respiratory problems and also became very deaf, which sadden him greatly since he could not enjoy music which had meant so much to him throughout his life. He was too ill in 1990 to attend the International Congress of Mathematicians in Kyoto. Friedrich Hirzebruch recalled in [12] his last meetings with Kodaira:-

Kunihiko Kodaira was friend and teacher for me. My wife and I remember our last visit to the Kodairas' house in Tokyo. He was working at the kitchen table on textbooks for secondary schools. Seiko Kodaira had to push the papers away when preparing the meal. In 1995 I congratulated him on his eightieth birthday. He answered in his charming way. But when we came to Tokyo in 1996, he was already in the hospital. We could not talk to him anymore.

Kodaira's wife, Seiko, died in January 2000, two and a half years after her husband.