Kunihiko Kodaira

Native spelling: 小平邦彦

1915–1997, Tokyo, Japan, Kōfu, Yamanashi Prefecture, Japan

GeometryTopologyAnalysisFields Medal 1954Wolf Prize in Mathematics 1984

Japanese mathematician — the first Japanese Fields Medallist — who applied harmonic integrals and sheaf cohomology to Kähler manifolds and algebraic varieties, proving the Kodaira vanishing and embedding theorems and classifying compact complex surfaces.

Kunihiko Kodaira graduated from the University of Tokyo in mathematics in 1938 and in physics in 1941. Working in wartime isolation, he mastered Hodge's theory of harmonic integrals and completed a 1949 doctoral thesis, 'Harmonic Fields in Riemannian Manifolds', that attracted the notice of Hermann Weyl and brought an invitation to the Institute for Advanced Study in Princeton.

At Princeton and the IAS during the 1950s, Kodaira combined harmonic forms with the new sheaf cohomology of Leray, Cartan, and Serre to prove the Kodaira vanishing theorem and the Kodaira embedding theorem (characterizing which compact Kähler manifolds are projective algebraic varieties), for which he received the Fields Medal in 1954. In a celebrated series of papers with Donald C. Spencer he then created the deformation theory of complex structures on compact manifolds.

In the 1960s Kodaira completed the classification of compact complex surfaces into rough classes, extending the classical Enriques–Italian classification beyond algebraic surfaces. After chairs at Johns Hopkins and Stanford he returned to the University of Tokyo in 1967, founding a prominent school of Japanese algebraic geometers, and was awarded the Wolf Prize in Mathematics in 1984.

Workplaces: University of Tokyo, Institute for Advanced Study, Princeton, Princeton University, Johns Hopkins University, Stanford University

Japan

Award-winning work

Fields Medal 1954Awarded for his extension of the theory of harmonic integrals to algebraic varieties. Building on Hodge's theory of harmonic forms, Kodaira used potential-theoretic and analytic methods to prove deep relations between the topology and the complex structure of Kähler manifolds, notably the Kodaira vanishing theorem for the cohomology of positive line bundles and the Kodaira embedding theorem characterizing projective algebraic manifolds. These results founded the modern transcendental approach to algebraic geometry and influenced the sheaf-cohomological work of Serre and Hirzebruch. He shared the 1954 Fields Medal with Jean-Pierre Serre.

Contributions, linked to the library

Library topicAlgebraic geometryProved the Kodaira vanishing and embedding theorems and classified compact complex surfaces.Library topicHomology and cohomologyApplied sheaf cohomology and harmonic forms to global problems in complex geometry.Library topicSmooth manifoldsWith Donald C. Spencer, founded the deformation theory of complex structures on compact manifolds.