# Heisuke Hironaka

Heisuke Hironaka ( Japanese 広 中 平 祐 , Hironaka Heisuke ; born April 9, 1931 in Yuu , Kuga-gun (today: Iwakuni ), Yamaguchi Prefecture , Japan ) is a Japanese mathematician and recipient of the Fields Medal .

## life and work

Hironaka was born as one of 15 children of a clothes dealer (and temporary textile manufacturer) in a town of 3000 people near Hiroshima . 1949-1954 he studied at the University of Kyoto initially physics , but then switched to mathematics that focus there under Yasuo Akizuki in abstract algebra was. In 1957 he accepted an invitation from Oscar Zariski , who had been in Kyoto the year before, to Harvard , where other leading algebraic geometers such as David Mumford , Steven Kleiman and Michael Artin studied at the same time . In 1959 he was invited by Alexander Grothendieck , with whom he became friends at Harvard in 1958/9, at the IHES in Paris. In his own words, his acquaintance with Grothendieck gave him essential impulses - a "global" perspective - for his later proof of the solvability of singularities. After completing his doctorate with Zariski in 1960, he went to Brandeis University , from 1964 to Columbia University in New York and from 1968 to Harvard as a professor . As early as 1975–1988 he was also a professor in Kyoto , where he was also director of the Research Institute for Mathematical Sciences (RIMS) from 1983–1985 . In Japan he is so respected and influential that his name is well known to many non-mathematicians. 1996–2002 he was director of Yamaguchi University in his home prefecture.

He worked in the field of algebraic geometry , as did the other two Fields medalists from Japan, Kunihiko Kodaira and Shigefumi Mori .

In 1964, Hironaka proved that the singularities of an algebraic variety of any dimension can be resolved over fields with the characteristic zero.

Before Hironaka, Robert Walker had already shown the solvability for algebraic surfaces over the complex numbers in 1935 after preliminary work by Giacomo Albanese et al. , Which dates back to the 19th century , and Zariski himself proved this in 1939 with purely algebraic methods for fields with the characteristic 0 (for Surfaces and curves). In 1944 he also proved that it can be resolved in characteristic 0 and dimension 3.

For his proof, which is almost 200 pages long and extremely difficult to understand, Hironaka received the Fields Medal in 1970 (lecture: Desingularization of complex analytic varieties ). The proof, which Hironaka himself does not consider to be complicated, has meanwhile been provided by Orlando Villamayor , Santiago Encinas , Edward Bierstone , Pierre Milman , Steven Dale Cutkosky , Herwig Hauser , János Kollár and others. a. has been simplified - it now fits on approx. 20 pages. De Jong gave an alternative proof with his method of alterations in 1997. Whether one can resolve singularities in positive characteristics (i.e. for varieties over finite fields) is only known in dimension 2, i.e. for algebraic surfaces (proof by SSAbhyankar 1956), but generally open to this day. Hironaka himself has been working on the proof for characteristic p until recently and published a preprint with an attempted proof on his website in 2017.

In 1962 he was invited speaker at the International Congress of Mathematicians in Stockholm ( On resolution of singularities (characteristic zero) ).

Hironaka is married to the politician Wakako Hironaka and has two children.

He was awarded the Asahi Prize in 1967 and elected to the American Academy of Arts and Sciences in 1969. In 1975 he was awarded the Japanese Order of Culture . In 1998 the asteroid (6978) Hironaka was named after him. Hironaka is an honorary doctorate from the University of Nice.

His autobiography (The Joy of Learning) is influential in Japan and South Korea (it appeared in Japanese and Korean).

## Quotes

The world is interesting because of its singularities ... You can look at smooth objects from a distance and see their shape, with singularities you have to get closer and closer .. Hawking said that in a black hole there is another universe. A singularity is something like this: if you look closely, you see a large universe. The problem with handling singularities is that they are only points, but contain a great many things. To see what's in it, you have to inflate it, enlarge it, smooth it out, then you can see the whole picture. (Hironaka, Interview, Notices AMS 2005)

## Works

• On the arithmetic genera and the effective genera of algebraic curves , Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math., Vol. 30, 1957, pp. 177-195.
• On the resolution of singularities (characteristic zero) , Proc. ICM Stockholm 1962, pp. 507-521
• An example of a non-Kählerian complex-analytic deformation of Kählerian complex structures , Ann. of Math. (2), Vol. 75, 1962, pp. 190-208.
• Resolution of singularities of an algebraic variety over a field of characteristic zero , Part I, II., Annals of Math. (2) Vol. 79, 1964, pp. 109-203; Pp. 205-326.
• with Hideyuki Matsumura Formal functions and formal imbeddings , J. Math. Soc. Japan, Vol. 20, 1968, pp. 52-82
• Additive groups associated with points of a projective space , Ann. of Math. (2), Vol. 92, 1970, pp. 327-334.
• On the characters and of singularities${\ displaystyle \ nu ^ {*}}$${\ displaystyle \ tau ^ {*}}$ , J. Math. Kyoto Univ., Volume 7, 1967, pp. 19-43
• Desingularization of complex-analytic varieties , Actes ICM, Nice 1970, Volume 2, pp. 627-631, Gauthier-Villars 1971
• Introduction to real-analytic sets and real-analytic maps , Quaderni dei Gruppi di Ricerca Matematica del Consiglio Nazionale delle Ricerche, Istituto Matematico “L. Tonelli ”dell'Università di Pisa, Pisa 1973
• Lectures on introduction to the theory of infinitely near singular points , Memorias de Matematica del Instituto “Jorge Juan”, No. 28, Madrid 1974
• with José M. Aroca, José L. Vicente The theory of the maximal contact , Memorias de Matematica del Instituto “Jorge Juan”, No. 29, Madrid 1975
• with Jose M. Aroca, Jose L. Vicente Desingularization theorems , Memorias de Matematica del Instituto “Jorge Juan”, No. 30, Madrid 1977
• with T. Urabe, Kaiseki Kukan Nyumon: Introduction to analytic spaces (Japanese), 2nd edition, Asakura Publ., Tokyo 1983
• Fame, Sweet and Bitter in Michael Atiyah u. a. Miscellanea Mathematica , Springer Verlag 1991
• Edited with Stanislaw Janeczko Geometric singularity theory , Warsaw 2004 (commemorative volume to Lojasiewicz )
• with José Manuel Aroc , José Luis Vicente : Complex Analytic Desingularization, Springer 2018

## Individual evidence

1. Interview, Notices AMS September 2005
2. Algebraic varieties are defined as the roots of polynomials over different fields. Bodies with characteristic zero are e.g. B. the complex or real numbers. In contrast, finite bodies have positive characteristics.
3. Partial results have e.g. B. Abhyankar found. In 1966 he proved the solvability in characteristic> 5 for dimension 3.
4. ^ American Academy of Arts and Sciences. Book of Members ( PDF ). Retrieved April 18, 2016
5. Minor Planet Circ. 32347