Kunihiko Kodaira
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Kunihiko Kodaira (小平 邦彦 Kodaira Kunihiko, 16 March, 1915 – 26 July, 1997) was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds; and as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954, being the first Japanese to receive this honour. He was born in Tokyo.
His early work was mostly in functional analysis. During the war years he worked in isolation, but was able to master Hodge theory as it then stood. He wrote a Ph.D. on it, finally presented in 1949; he had been involved in cryptographic work from about 1944, at a time of great personal difficulty, while holding an academic post in Tokyo.
In 1949 he travelled to the IAS in Princeton, at the invitation of Hermann Weyl. At this time the foundations of Hodge theory were being brought in line with contemporary technique in operator theory. Kodaira rapidly became involved in exploiting the tools it opened up in algebraic geometry, adding sheaf theory as it became available. This work was particularly influential, for example on Hirzebruch.
In a second research phase, Kodaira wrote a long series of papers in collaboration with D. C. Spencer, founding the deformation theory of complex structures on manifolds. This gave the possibility of constructions of moduli spaces, since in general such structures depend continuously on parameters. It also identified the sheaf cohomology groups, for the sheaf associated with the holomorphic tangent bundle, that carried the basic data about the dimension of the moduli space, and obstructions to deformations. This theory is still foundational, and also had an influence of the (technically very different) scheme theory of Grothendieck. Spencer then continued this work, applying the techniques to structures other than complex ones, such as Gstructures.
In a third major part of his work, Kodaira worked again from around 1960 through the classification of algebraic surfaces, from birational geometry, from the point of view of complex manifold theory. This resulted in a typology of seven kinds of twodimensional compact complex manifolds, recovering the five algebraic types known classically; the other two being nonalgebraic. He provided also detailed studies of elliptic fibrations of surfaces over a curve, or in other language elliptic curves over function fields, a theory whose arithmetic analogue proved important soon afterwards. This work also included a characterisation of K3 surfaces as deformations of quartic surfaces in P^{4}, and the theorem that they form a single diffeomorphism class. Again, this work has proved foundational.
Kodaira left the IAS in 1961, and after two positions in the USA returned to Japan in 1967; he was professor at the University of Tokyo. He was awarded a Wolf Prize in 1984/5. He died at Kofu.
See also:
 TitchmarshKodaira formula
 Kodaira vanishing theorem
 KodairaSpencer mapping
 Kodaira dimension
 Kodaira embedding theorem
External link
MacTutor biography (http://wwwhistory.mcs.standrews.ac.uk/history/Mathematicians/Kodaira.html)de:Kunihiko Kodaira ja:小平邦彦 sv:Kunihiko Kodaira