## Kunihiko Kodaira (Fields Medal 1954)

© Graduate School of Mathematical Sciences, The University of Tokyo

The late Prof. Kunihiko Kodaira developed the theory of complex manifolds. The breadth and depth of his research exerted a great impact on the fields of algebraic geometry, complex analysis and mathematical physics. He decided to pursue a career in research during the Second World War. In 1949, not long after the war, he was invited to the Institute for Advanced Study at Princeton as the second Japanese mathematician in the post-war period to work in the United States (the first was Shizuo Kakutani). Ater that, he wrote 50 papers (1,400 pages in total) during his 19-year stay in the United States.

Complex manifolds are high-dimensional geometric objects that have complex numbers as coordinates. They are invisible to the naked eye except for Riemann surfaces, which are one-dimensional complex manifolds. But by using analytical methods such as harmonic integrals and/or algebraic machinery such as sheaf cohomology, Prof. Kodaira found that one could develop a geometry of complex manifolds as rich as that of visible figures, a discovery of great importance. A cornerstone of Prof. Kodaira's work was the theory of harmonic forms, which he studied during the war years, and submitted as the topic of a doctoral thesis in the period of confusion after the war. This dissertation was too lengthy to be published in Japan due to the paper shortage, and Prof. Kodaira asked Prof. Kakutani to take a typewritten manuscript to have it published in the United States. The article attracted the attention of the great mathematician Hermann Weyl, who immediately decided to invite Prof. Kodaira to Princeton.

Prof. Kodaira's major achievements include the Kodaira vanishing theorem, deformation theory of complex structure, and the classification of complex surfaces. He was the first Japanese to receive the Fields Medal in 1954 for his series of work on harmonic analysis represented by the Kodaira vanishing theorem. Deformation theory, the result of joint research with his friend D.C. Spencer, was awarded the Japan Academy Prize. Prof. Kodaira's theory of complex surfaces provided a rigorous foundation for the Enriques classification of algebraic surfaces, which originated in Italy in the first half of the 20th century, extended it to a larger category, and provided a good model for the current high-dimensional algebraic geometry.

Prof. Kodaira supervised more than a dozen excellent students during the seven years at the University of Tokyo after his return to Japan in 1968. His students were called "the Kodaira School," and they were compared to "the Nagata School," who were students of Prof. Masayoshi Nagata at Kyoto University. These two schools together made Japan a global research center for algebraic geometry and complex manifold theory.

As a child, Prof. Kodaira did not like physical education, especially running, but he had an extraordinary interest in numbers and geometry. There is a funny story. One day, he decided to conduct an experiment to see if animals could understand numbers. He took three puppies from their mother and returned only two of them. When he saw the mother dog was not aware of the absence of one puppy, he concluded that dogs do not have the concept of numbers. Later in his life, Prof. Kodaira evinced an interest in mathematics education, and emphasized that education should follow children's developmental stages. He contended that elementary school education should focus on language skills and mathematics.

In his spare time, he enjoyed music and was an accomplished piano player. He became acquainted with his wife when he accompanied her violin performance on the piano. The first piano he bought in the United States was second hand and half a tone lower than normal, and he finally became able to play any musical pieces by transposing up a half tone.

See also: http://www.mathunion.org/general/prizes/fields/details/

### Career timeline

- Born March 16, 1915 (in Yodobashi, Tokyo)
- Died on July 26, 1997 at the age of 82

#### Academic history and professional highlights

March 1938 | Bachelor of science in mathematics, Faculty of Science, the Imperial University of Tokyo |
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March 1941 | Bachelor of science in physics, Faculty of Science, the Imperial University of Tokyo |

April 1941 | Lecturer, Department of Physics, Faculty of Science, the Imperial University of Tokyo |

April 1942 | Associate professor, Department of Mathematics, Faculty of Science, Tokyo Bunri University |

April 1944 | Associate professor, Department of Physics, Faculty of Science, the Imperial University of Tokyo (concurrent position) |

April 1949 | PhD, Graduate School of Science, the University of Tokyo |

September 1949 | Fellow, Institute for Advanced Study, Princeton |

September 1950 | Visiting professor, Johns Hopkins University |

June 1951 | Fellow, Institute for Advanced Study, Princeton |

September 1952 | Associate professor, Princeton University |

September 1955 |
Professor, Princeton University Fellow, Institute for Advanced Study, Princeton (Resignation as professor from the University of Tokyo) |

September 1961 - 1962 | Visiting professor, Harvard University |

September 1962 | Professor, Johns Hopkins University |

September 1965 | Professor, Stanford University |

November 1965 | Member, the Academy of Japan |

September 1968 | Professor, Department of Mathematics, Faculty of Science, the University of Tokyo |

April 1971- 1973 | Dean, Faculty of Science, the University of Tokyo |

March 1975 | Retirement from the University of Tokyo |

April 1975 | Professor, Faculty of Science, Gakushuin University |

May 1978 | Foreign correspondent member, American Academy of Arts and Sciences |

March 1985 | Retirement from Gakushuin University |

1986 – 1990 | Chair, International Congress of Mathematicians |

#### Honors & Awards

1954 | Fields Medal |
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1957 | The Japan Academy Prize (Japan Academy), The Order of Culture (Government of Japan) |

1984 | Wolf Prize |