##
**An invitation to Kuranishi mathematics.
(Kuranishi sūgaku eno izanai.)**
*(Japanese)*
Zbl 1337.32002

Tokyo: Iwanamishoten (ISBN 978-4-00-005272-6). 189 p. (2013).

This book, whose subtitle is “Masatake Kuranishi: His Life and Mathematics”, is a companion volume of M. Kuranishi [Selected Papers of Masatake Kuranishi. Edited by Takao Akahori, Gen Komatsu, Kimio Miyajima, Makoto Namba, Duong H. Phong and Keizo Yamaguchi. Hackensack, NJ: World Scientific (2013; Zbl 1295.32001)], though it is written in Japanese. The book is divided into two parts. The first part is concerned with Kuranishi as a person, and the second part deals with mathematical works of Kuranishi.

The first part consists of roughly 100 pages, about four fifths of which is his autobiography based on several interviews to Kuranishi by Tadashi Tomaru. The first part contains also three essays, namely, Memories of Kuranishi written by Victor Guillemin (his English essay is followed by its Japanese translation), Memories of Kuranishi written by Kuranishi’s younger brother (Shigeru Kuranishi) and Memories of an apprentice under Professor Kuranishi written by Makoto Namba.

The second part consists of five reviews on the mathematics of Kuranishi. The first review, written by Tohru Morimoto, is concerned with geometric theory of partial differential equations centering M. Kuranishi’s [Am. J. Math. 79, 1–47 (1957; Zbl 0077.29701); erratum ibid. 79, 448 (1957); Nagoya Math. J. 15, 225–260 (1961; Zbl 0212.56501); Lectures on involutive systems of partial differential equations. Paulo: Publicacoes da Sociedade de Matematica de Sao Paulo (1967; Zbl 0163.12001)]. The second review, written by Akira Fujiki, is concerned with Kuranishi families in deformations of compact complex manifolds centering M. Kuranishi’s [Ann. Math. (2) 75, 536–577 (1962; Zbl 0106.15303); in: Proc. Conf. Complex Analysis, Minneapolis 1964, 142–154 (1965; Zbl 0144.21102); in: Global Analysis, Papers in Honor of K. Kodaira 309–313 (1969; Zbl 0211.10301); Deformations of compact complex manifolds. Montreal, Canada: Les Presses de l’Universite de Montreal (1971; Zbl 0256.32014)]. The third review, written by Kimio Miyajima, is concerned with CR manifolds centering M. Kuranishi’s [Ann. Math. (2) 115, 451–500 (1982; Zbl 0505.32018); ibid. 116, 1–64 (1982; Zbl 0505.32019); ibid. 116, 249–330 (1982; Zbl 0576.32033)]. The fourth review, written by Mitsuhiro Ito, is concerned with Yang-Mills connections and Kuranishi mappings centering M. Kuranishi’s [in: Proc. Conf. Complex Analysis, Minneapolis 1964, 142–154 (1965; Zbl 0144.21102)]. The fifth review, written by Ryushi Goto, is concerned with generalized complex structures and their deformation theory centering M. Kuranishi’s [Ann. Math. (2) 75, 536–577 (1962; Zbl 0106.15303); in: Proc. Conf. Complex Analysis, Minneapolis 1964, 142–154 (1965; Zbl 0144.21102)].

Kuranishi was born in Tokyo in 1924, when Tokyo was still in a turmoil after the Great Kanto Earthquake in 1923. He entered Nagoya University in 1944, when the situation in the Pacific War was deteriorating day by day to Japan. Nagoya University was founded in 1939, when the Second World War erupted. Then and there he met a number of brilliant professors, say, Kosaku Yoshida, Tadashi Nakayama, Yozo Matsushima, Kiyoshi Ito and Goro Azumaya. We can find Noboru Ito and Nobuo Shimada among his peers, the first being destined to become famous in the theory of finite groups and the second being specialized in algebraic topology. After graduation, Kuranishi became an instructor of Tokyo Institute of Technology, where his first paper [H. Toyama and M. Kuranishi, Kōdai Math. Semin. Rep. 1949, 17–18 (1949; Zbl 0054.01601)] was written and he had spent three years until he moved to Nagoya University. Until 1952, when he got his Ph.D. from Nagoya University, David Hilbert’s fifth problem concerning the characterization of Lie groups had occupied a central position in the mind of Kuranishi. With respect to this, he has written two papers [Proc. Am. Math. Soc. 1, 372–380 (1950; Zbl 0038.01701); Nagoya Math. J. 1, 71–81 (1950; Zbl 0037.30502)], which have contributed greatly to [H. Yamabe, Ann. Math. (2) 58, 351–365 (1953; Zbl 0053.01602)].

As is well known, it is not easy to read publications of Élie Cartan, though they are all significant contributions to mathematics. It was Yozo Matsushima who invited Kuranishi to mathematics of Élie Cartan. Kuranishi learned from Élie Cartan that the first step in the study of some mathematical structure should be the thorough study of a good model, and the structure itself should be understood as a deformation of the model. The use of differential forms in Kuranishi’s later study of complex structures and CR structures is to be attributed to his encounter with publications of Élie Cartan, who has founded the theory of differential forms. Since Hilbert’s fifth problem was settled, Kuranishi’s main interest was then oriented towards geometric theory of partial differential equations. Élie Cartan is known to have devoted all his energy to the study of Pfaff systems or exterior differential systems and pseudogroups in the first decade of the 20th century (E. Cartan [JFM 30.0313.04], [JFM 32.0351.04], [JFM 32.0351.05], [JFM 33.0351.01], [JFM 33.0356.01], [JFM 41.0417.01], [JFM 35.0176.04], [JFM 40.0193.02] and so on). M. Kuranishi’s first work in this area is [Am. J. Math. 79, 1–47 (1957; Zbl 0077.29701); erratum ibid. 79, 448 (1957)], which was to play a pivotal role in his study of deformations of complex structures, and which Kuranishi considers one of his most important and most fundamental works. M. Kuranishi’s work with respect to pseudogroups is [Nagoya Math. J. 15, 225–260 (1961; Zbl 0212.56501)].

Thanks to D. Montgomery’s invitation, Kuranishi was entitled to spend two years since 1954 at Institute for Advanced Study. He then spent a year and a half at Chicago University, where he met A. Weil, S. S. Chern, A. P. Calderón and A. Zygmund, and at Massachusetts Institute of Technology. Calderón and Zygmund are famous for the theory of singular integrals (A. Zygmund [Rend. Mat. Appl., V. Ser. 16, 468–505 (1958; Zbl 0088.08302)], A. P. Calderón and A. Zygmund [Am. J. Math. 79, 901–921 (1957; Zbl 0081.33502); Am. J. Math. 78, 310–320 (1956; Zbl 0072.11601); Am. J. Math. 78, 289–309 (1956; Zbl 0072.11501)]), which was developed into the theory of pseudo-differential operators by Joseph J. Kohn, Lars Hörmander and Louis Nirenberg in the 1960s. In 1961 Kuranishi spent three months at the Tata Institute of Fundamental Research, where C. L. Segal stayed at that time, Kuranishi happened to meet Henri Cartan, whose father is Élie Cartan, and Kuranishi gave a lecture entitled “On Exterior Differential Systems”, whose lecture notes by Venkatesha Murthy were published there.

Kuranishi’s intimate friendship with K. Kodaira began in 1954, when Kuranishi stayed at IAS. Kodaira and D. C. Spencer are famous for the deformation theory of complex structures ([Ann. Math. (2) 76, 306–398, 399–445 (1962; Zbl 0124.38601); ibid. 81, 389–450 (1965; Zbl 0192.29603)]), and M. Kuranishi’s first contribution in this area is [ibid. 74, 262–328 (1961; Zbl 0192.18501)], which enticed Kodaira and Spencer to invite Kuranishi to Princeton University as a research fellow for a year since September 1960. Besides Kodaira and Spencer, S. Lefschetz, E. Artin and S. Bochner were enrolled there at that time. The Kodaira-Spencer seminar at Princeton University, of which R. Gunning was a regular member, inspired M. Kuranishi to finish the paper [ibid. 75, 536–577 (1962; Zbl 0106.15303)]. In September 1961 Kuranishi moved from Nagoya University to Columbia University, where he had stayed until he retired at the age of 75 in 1999. V. W. Guillemin was once an instructor at Columbia University, and has written [Am. J. Math. 90, 1307–1320 (1968; Zbl 0186.16403)] with M. Kuranishi. M. Namba stayed at Columbia University as a foreign student for three years and a half in the 1960s. In New York, Louis Nirenberg, younger than Kuranishi by a year, lived near Kuranishi. Among colleagues of Kuranishi at Columbia University we can find Samuel Eilenberg, who is famous for his successful books (S. Eilenberg and N. Steenrod [Zbl 0047.41402] and H. Cartan and S. Eilenberg [Zbl 0075.24305]) and had once Daniel Kan, William Lawvere and K. T. Chen among his students, and also Richard Hamilton, who is famous for [R. Hamilton, Zbl 0504.53034] leading to [G. Perelman, Zbl 1130.53001; Zbl 1130.53002].

You can find more information in the book, and the reviewer urges strongly that the book should be translated into English.

The first part consists of roughly 100 pages, about four fifths of which is his autobiography based on several interviews to Kuranishi by Tadashi Tomaru. The first part contains also three essays, namely, Memories of Kuranishi written by Victor Guillemin (his English essay is followed by its Japanese translation), Memories of Kuranishi written by Kuranishi’s younger brother (Shigeru Kuranishi) and Memories of an apprentice under Professor Kuranishi written by Makoto Namba.

The second part consists of five reviews on the mathematics of Kuranishi. The first review, written by Tohru Morimoto, is concerned with geometric theory of partial differential equations centering M. Kuranishi’s [Am. J. Math. 79, 1–47 (1957; Zbl 0077.29701); erratum ibid. 79, 448 (1957); Nagoya Math. J. 15, 225–260 (1961; Zbl 0212.56501); Lectures on involutive systems of partial differential equations. Paulo: Publicacoes da Sociedade de Matematica de Sao Paulo (1967; Zbl 0163.12001)]. The second review, written by Akira Fujiki, is concerned with Kuranishi families in deformations of compact complex manifolds centering M. Kuranishi’s [Ann. Math. (2) 75, 536–577 (1962; Zbl 0106.15303); in: Proc. Conf. Complex Analysis, Minneapolis 1964, 142–154 (1965; Zbl 0144.21102); in: Global Analysis, Papers in Honor of K. Kodaira 309–313 (1969; Zbl 0211.10301); Deformations of compact complex manifolds. Montreal, Canada: Les Presses de l’Universite de Montreal (1971; Zbl 0256.32014)]. The third review, written by Kimio Miyajima, is concerned with CR manifolds centering M. Kuranishi’s [Ann. Math. (2) 115, 451–500 (1982; Zbl 0505.32018); ibid. 116, 1–64 (1982; Zbl 0505.32019); ibid. 116, 249–330 (1982; Zbl 0576.32033)]. The fourth review, written by Mitsuhiro Ito, is concerned with Yang-Mills connections and Kuranishi mappings centering M. Kuranishi’s [in: Proc. Conf. Complex Analysis, Minneapolis 1964, 142–154 (1965; Zbl 0144.21102)]. The fifth review, written by Ryushi Goto, is concerned with generalized complex structures and their deformation theory centering M. Kuranishi’s [Ann. Math. (2) 75, 536–577 (1962; Zbl 0106.15303); in: Proc. Conf. Complex Analysis, Minneapolis 1964, 142–154 (1965; Zbl 0144.21102)].

Kuranishi was born in Tokyo in 1924, when Tokyo was still in a turmoil after the Great Kanto Earthquake in 1923. He entered Nagoya University in 1944, when the situation in the Pacific War was deteriorating day by day to Japan. Nagoya University was founded in 1939, when the Second World War erupted. Then and there he met a number of brilliant professors, say, Kosaku Yoshida, Tadashi Nakayama, Yozo Matsushima, Kiyoshi Ito and Goro Azumaya. We can find Noboru Ito and Nobuo Shimada among his peers, the first being destined to become famous in the theory of finite groups and the second being specialized in algebraic topology. After graduation, Kuranishi became an instructor of Tokyo Institute of Technology, where his first paper [H. Toyama and M. Kuranishi, Kōdai Math. Semin. Rep. 1949, 17–18 (1949; Zbl 0054.01601)] was written and he had spent three years until he moved to Nagoya University. Until 1952, when he got his Ph.D. from Nagoya University, David Hilbert’s fifth problem concerning the characterization of Lie groups had occupied a central position in the mind of Kuranishi. With respect to this, he has written two papers [Proc. Am. Math. Soc. 1, 372–380 (1950; Zbl 0038.01701); Nagoya Math. J. 1, 71–81 (1950; Zbl 0037.30502)], which have contributed greatly to [H. Yamabe, Ann. Math. (2) 58, 351–365 (1953; Zbl 0053.01602)].

As is well known, it is not easy to read publications of Élie Cartan, though they are all significant contributions to mathematics. It was Yozo Matsushima who invited Kuranishi to mathematics of Élie Cartan. Kuranishi learned from Élie Cartan that the first step in the study of some mathematical structure should be the thorough study of a good model, and the structure itself should be understood as a deformation of the model. The use of differential forms in Kuranishi’s later study of complex structures and CR structures is to be attributed to his encounter with publications of Élie Cartan, who has founded the theory of differential forms. Since Hilbert’s fifth problem was settled, Kuranishi’s main interest was then oriented towards geometric theory of partial differential equations. Élie Cartan is known to have devoted all his energy to the study of Pfaff systems or exterior differential systems and pseudogroups in the first decade of the 20th century (E. Cartan [JFM 30.0313.04], [JFM 32.0351.04], [JFM 32.0351.05], [JFM 33.0351.01], [JFM 33.0356.01], [JFM 41.0417.01], [JFM 35.0176.04], [JFM 40.0193.02] and so on). M. Kuranishi’s first work in this area is [Am. J. Math. 79, 1–47 (1957; Zbl 0077.29701); erratum ibid. 79, 448 (1957)], which was to play a pivotal role in his study of deformations of complex structures, and which Kuranishi considers one of his most important and most fundamental works. M. Kuranishi’s work with respect to pseudogroups is [Nagoya Math. J. 15, 225–260 (1961; Zbl 0212.56501)].

Thanks to D. Montgomery’s invitation, Kuranishi was entitled to spend two years since 1954 at Institute for Advanced Study. He then spent a year and a half at Chicago University, where he met A. Weil, S. S. Chern, A. P. Calderón and A. Zygmund, and at Massachusetts Institute of Technology. Calderón and Zygmund are famous for the theory of singular integrals (A. Zygmund [Rend. Mat. Appl., V. Ser. 16, 468–505 (1958; Zbl 0088.08302)], A. P. Calderón and A. Zygmund [Am. J. Math. 79, 901–921 (1957; Zbl 0081.33502); Am. J. Math. 78, 310–320 (1956; Zbl 0072.11601); Am. J. Math. 78, 289–309 (1956; Zbl 0072.11501)]), which was developed into the theory of pseudo-differential operators by Joseph J. Kohn, Lars Hörmander and Louis Nirenberg in the 1960s. In 1961 Kuranishi spent three months at the Tata Institute of Fundamental Research, where C. L. Segal stayed at that time, Kuranishi happened to meet Henri Cartan, whose father is Élie Cartan, and Kuranishi gave a lecture entitled “On Exterior Differential Systems”, whose lecture notes by Venkatesha Murthy were published there.

Kuranishi’s intimate friendship with K. Kodaira began in 1954, when Kuranishi stayed at IAS. Kodaira and D. C. Spencer are famous for the deformation theory of complex structures ([Ann. Math. (2) 76, 306–398, 399–445 (1962; Zbl 0124.38601); ibid. 81, 389–450 (1965; Zbl 0192.29603)]), and M. Kuranishi’s first contribution in this area is [ibid. 74, 262–328 (1961; Zbl 0192.18501)], which enticed Kodaira and Spencer to invite Kuranishi to Princeton University as a research fellow for a year since September 1960. Besides Kodaira and Spencer, S. Lefschetz, E. Artin and S. Bochner were enrolled there at that time. The Kodaira-Spencer seminar at Princeton University, of which R. Gunning was a regular member, inspired M. Kuranishi to finish the paper [ibid. 75, 536–577 (1962; Zbl 0106.15303)]. In September 1961 Kuranishi moved from Nagoya University to Columbia University, where he had stayed until he retired at the age of 75 in 1999. V. W. Guillemin was once an instructor at Columbia University, and has written [Am. J. Math. 90, 1307–1320 (1968; Zbl 0186.16403)] with M. Kuranishi. M. Namba stayed at Columbia University as a foreign student for three years and a half in the 1960s. In New York, Louis Nirenberg, younger than Kuranishi by a year, lived near Kuranishi. Among colleagues of Kuranishi at Columbia University we can find Samuel Eilenberg, who is famous for his successful books (S. Eilenberg and N. Steenrod [Zbl 0047.41402] and H. Cartan and S. Eilenberg [Zbl 0075.24305]) and had once Daniel Kan, William Lawvere and K. T. Chen among his students, and also Richard Hamilton, who is famous for [R. Hamilton, Zbl 0504.53034] leading to [G. Perelman, Zbl 1130.53001; Zbl 1130.53002].

You can find more information in the book, and the reviewer urges strongly that the book should be translated into English.

Reviewer: Hirokazu Nishimura (Tsukuba)

### MSC:

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

32-03 | History of several complex variables and analytic spaces |

53-03 | History of differential geometry |

01A60 | History of mathematics in the 20th century |