Remembering Shoshichi Kobayashi Gary R. Jensen, Coordinating Editor hoshichi Kobayashi was born January 4, 1932, in Kofu City, Japan. He grew up amidst the colossal devastation of World War II. After completing a BS degree Sin 1953 at the University of Tokyo, he spent the academic year 1953–1954 on a French scholarship at the University of Paris and the University of Strasbourg. From there he went to the University of Washington in Seattle, where he received a Ph.D. in 1956 for the thesis “Theory of connections,” (Annali di Mat. (1957), 119–194), written under the direction of Carl B. Allendoerfer. There followed two years as a member of the Photo from the Oberwolfachcourtesy Photo of Collection, the Archives of the MFO. Institute for Advanced Study in Princeton and then Shoshichi Kobayashi two years as a research associate at MIT. In 1960 he became an assistant professor at the University of British Columbia, which he left in 1962 to go to many of Shoshichi’s mathematical achievements. the University of California at Berkeley, where he Several of his students relate their experiences of remained for the rest of his life. working under Professor Kobayashi’s supervision. Shoshichi published 134 papers, widely admired Some Berkeley colleagues write of their personal not only for their mathematical content but for the experiences with Shoshichi. elegance of his writing. Among his thirteen books, His brother, Hisashi, recounts some details of Hyperbolic Manifolds and Holomorphic Mappings Shoshichi’s early education. Shoshichi’s wife and and the Kobayashi metric in complex manifolds daughters tell a fascinating story of his childhood, created a field, while Foundations of Differential his studies abroad, and major life experiences Geometry, I and II, coauthored with Nomizu, are preceding his move to Berkeley. known by virtually every differential geometry He died August 29, 2012, of heart failure on a student of the past fifty years. In their articles flight from Tokyo to San Francisco. below, Shoshichi’s student, Toshiki Mabuchi at Osaka University and former Professor Takushiro Toshiki Mabuchi Ochiai at the University of Tokyo summarize Many of Professor Kobayashi’s books are known as standard references in differential geometry, com- Gary R. Jensen is professor emeritus of mathematics at plex geometry, and other related areas. Especially, Washington University in St. Louis. His email address is Foundations of Differential Geometry, Vols. I and II, [email protected]. Personal and family photos of S. Kobayashi provided by and Toshiki Mabuchi is professor of mathematics at Osaka used with permission of the Nomizu family. University. His email address is [email protected]. DOI: http://dx.doi.org/10.1090/noti1184 osaka-u.ac.jp. 1322 Notices of the AMS Volume 61, Number 11 coauthored by Nomizu, are very popular with Takushiro Ochiai 1 mathematicians as well as with physicists. He The life and academic achievements of Shoshichi and Nomizu received the 2007 MSJ Publication Kobayashi give a definition of what a great Prize from the Mathematical Society of Japan for mathematician should be. He left us numerous this work. His books Hyperbolic Manifolds and manuscripts of high originality, comparable to Holomorphic Mappings (1970) and Transformation great musical compositions. His books, taken as Groups in Differential Geometry (1972) also influ- a whole, harmonize splendidly into a great sym- enced many mathematicians. His mathematical phony. Though aware of my inability to reach the achievements range across differential geometry, height of his talent, I dare to write this article to Lie algebras, transformation groups, and complex bring attention to the fine personal character and analysis. The most important ones are: remarkable academic achievements of Shoshichi 1. the Kobayashi intrinsic pseudo-distance, Kobayashi, outstanding mathematician, mentor, and colleague. 2. the Kobayashi hyperbolicity and measure Kobayashi published papers in academic jour- hyperbolicity, nals every year without fail, starting from his virgin 3. projectively invariant distances for affine and paper in 1954 until his last days. Among a total projective distances, of 134 papers, there are 85 single-author papers 4. the study of compact complex manifolds with and 49 collaborative works. A unique feature positive Ricci curvature, of his single-author publications is their brevity. 5. filtered Lie algebras and geometric structures, There are 34 papers of less than five pages, 28 and papers of less than ten pages, and 15 papers of 6. Kobayashi-Hitchin correspondence for vector less than 28 pages. Every one of them is deep bundles. and rich in content with transparent and easily In (1) and (2) we see his outstanding creativ- comprehensible explanations, as is evidenced by ity. Kobayashi’s distance decreasing property for his invention of the concepts of the Kobayashi holomorphic mappings plays a very important role distance, hyperbolic complex manifolds, and the in (1), while the generalized Schwarz lemma is Hermitian-Einstein vector bundles. Inspired by Chern’s result, which further improved on the crucially used in (2). His works now give us a funda- generalization of the classical Schwarz’s lemma mental tool in the study of holomorphic mappings by L. Ahlfors (“The holomorphic maps between between complex manifolds. For instance, Picard’s Hermitian manifolds” by S. S. Chern), Kobayashi small theorem follows easily from (2). Recently, became interested in Schwarz’s lemma and read the above results were being generalized by him all the related papers one by one. He especially to almost complex manifolds. admired Carathéodory’s point of view and devoted On the other hand, (4) has led succeeding mathe- himself to and took advantage of the holomorphic maticians to Frankel’s Conjecture and Hartshorne’s maps, which he admitted led him to the discovery Conjecture. Among them, his joint work with T. of the Kobayashi distance. Ochiai, “Characterization of complex projective Expanding all his papers, paying attention to the spaces and hyperquadrics,” J. Math. Kyoto U. 13 historic background and development, Kobayashi (1972), 31–47, was effectively used in Siu-Yau’s subsequently published thirteen self-contained proof of Frankel’s Conjecture. It should also be books, every one of which is easily comprehensible noted that the method of reduction modulo p in by graduate students. Professor Kobayashi once said to me: “When I write books, I prepare one Mori’s proof of Hartshorne’s Conjecture became or two years for the first manuscript and make a clue to the Mori theory on the minimal model certain to give lectures based on it for one or two program of projective algebraic manifolds. years in order to deepen the contents before finally The Kobayashi-Hitchin correspondence for vec- completing the final manuscript.” tor bundles in (6) states that a holomorphic vector To convey to the public his philosophy of mathe- bundle E over a compact Kähler manifold is stable matics, or rather of differential geometry, I include in the sense of Mumford-Takemoto if and only if here passages from his essays in Japanese as pub- E admits a Hermitian-Einstein metric. Kobayashi lished in well-known magazines on mathematical and Lübke proved the “if” part, while the “only if” part, conjectured by Kobayashi and Hitchin, was Takushiro Ochiai is professor emeritus of mathematics at the University of Tokyo. His email address is ochia@ proved finally by Donaldson and Uhlenbeck-Yau. nittai.ac.jp. Since 1995 Professor Kobayashi regularly at- 1This is a translation into English of some part of my article tended our annual workshop on complex geometry “Kobayashi-sensei wo shinonde,” which appeared in S¯ugaku at Sugadaira, Japan. Seminar, 2013. December 2014 Notices of the AMS 1323 “Some percentage of the work in mathemat- ics consists of the creation of suitable notation. Suitable notation makes calculations easy to han- dle, makes formulas look beautiful and easy to memorize, and makes theorems apparent at a glance. Suitable notation is important not only from a passive role to make descriptions beautiful and easy, but also from an active role, even to suggesting what should be done next. We cannot explain in one word what kind of notation is suitable or what kind of theorem is good. It is a mathematical sense that will take it in somehow.” (S¯ugakuSeminar, September 1967) First row: T. Ochiai, S. Kobayashi, S.-T. Yau; “When you study subjects in mathematics which Second row: H. Mori, S.-S. Chern. Tokyo, 1977. are not restricted to calculus, I would like to urge you to study the history of its development sciences which deeply reflect his mathematical together by all means. Since modern mathematics achievements. The transparency, beauty, and open- requires too much stringency, lectures as well mindedness of his viewpoints, which he sharpened as textbooks rarely touch historical background. well to the limit, express my personal and eternal However, I think that one can understand the yearning. subject more deeply by getting to know the history “Differential geometry is one viewpoint over and ‘why this concept was produced.’ I want people mathematics, and a method, in itself. When various who become school teachers to study the history fields in mathematics reach the point of being well of mathematics by all means.” (Suuri Kagaku, understood, the phenomenon called ‘algebraiza- February 2001) tion’ occurs. I think it is possible to see different Through my acquaintance with Professor fields of mathematics from a differential geometric Kobayashi for many years, I am convinced that viewpoint as well as from an algebraic viewpoint. his passion for mathematics was motivated by The raison d’etre of differential geometry is to his love for human beings and for scholarship. I offer a new viewpoint and powerful methods rather am extremely fortunate to have collaborated with than being considered like a theory of numbers. Professor Kobayashi on some of his mathematical Moreover, the concepts and methods understood work.