**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 year1949-1950. 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. The1950s 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. In1995I congratulated him on his eightieth birthday. He answered in his charming way. But when we came to Tokyo in1996, 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.

**Article by:** *J J O'Connor* and *E F Robertson*

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