Heisuke Hironaka

Native spelling: 広中平祐

fl. 1931, Yuu-chō (present-day Iwakuni), Yamaguchi Prefecture, Japan

AlgebraGeometryFields Medal 1970

Japanese mathematician who proved the resolution of singularities of algebraic varieties in characteristic zero, a cornerstone of modern algebraic geometry; Fields Medal 1970.

Heisuke Hironaka was born in Yamaguchi Prefecture, Japan, into a large family, and initially studied physics at Kyoto University before turning to mathematics, earning his degree there under Yasuo Akizuki. In 1957 he went to Harvard University to work with Oscar Zariski, receiving his PhD in 1960.

At Harvard, Hironaka carried out his monumental proof that singularities of algebraic varieties over fields of characteristic zero can be resolved by a finite sequence of blow-ups, settling a question that had been open since the work of Zariski on surfaces and threefolds. The two-part paper, published in the Annals of Mathematics in 1964, introduced idealistic exponents and a tower of induction that made the result provable in arbitrary dimension; the corresponding problem in positive characteristic remains unsolved in full generality.

Hironaka taught at Brandeis University, Columbia University and Harvard before returning to Japan in 1975 as a professor at Kyoto University, where he later directed the Research Institute for Mathematical Sciences. He received the Fields Medal in 1970. Beyond research he is known in Japan as an essayist and as a promoter of mathematical education.

Workplaces: Harvard University, Columbia University, Kyoto University

Japan, United States

Award-winning work

Fields Medal 1970Awarded for his work on algebraic varieties, in particular his 1964 proof of the resolution of singularities for algebraic varieties in characteristic zero: every such variety can be transformed into a smooth one by a finite sequence of blow-ups along smooth centers lying in its singular locus. The proof introduced subtle induction invariants — orders of ideals under local blow-up — whose monotonization controls the process and which remain the basis of all known proofs. Resolution of singularities became a foundational tool of modern algebraic geometry, used for desingularizing moduli spaces, compactifying problems and computing invariants.

Contributions, linked to the library

Library topicAlgebraic geometryProved resolution of singularities in characteristic zero, the central existence theorem of birational algebraic geometry.