by Toshiki Mabuchi
Many of Professor Kobayashi’s books are known as standard references in differential geometry, complex geometry, and other related areas. Especially, Foundations of Differential Geometry, vols. I and II, coauthored by Nomizu, are very popular with mathematicians as well as with physicists. He and Nomizu received the 2007 MSJ Publication Prize from the Mathematical Society of Japan for this work. His books Hyperbolic Manifolds and Holomorphic Mappings (1970) and Transformation Groups in Differential Geometry (1972) also influenced many mathematicians. His mathematical achievements range across differential geometry, Lie algebras, transformation groups, and complex analysis. The most important ones are:
- the Kobayashi intrinsic pseudo-distance,
- the Kobayashi hyperbolicity and measure hyperbolicity,
- projectively invariant distances for affine and projective distances,
- the study of compact complex manifolds with positive Ricci curvature,
- filtered Lie algebras and geometric structures, and
- Kobayashi–Hitchin correspondence for vector bundles.
In (1) and (2) we see his outstanding creativity. Kobayashi’s distance decreasing property for holomorphic mappings plays a very important role in (1), while the generalized Schwarz lemma is crucially used in (2). His works now give us a fundamental tool in the study of holomorphic mappings between complex manifolds. For instance, Picard’s small theorem follows easily from (2). Recently, the above results were being generalized by him to almost complex manifolds.
On the other hand, (4) has led succeeding mathematicians to Frankel’s Conjecture and Hartshorne’s Conjecture. Among them, his joint work with T. Ochiai, “Characterization of complex projective spaces and hyperquadrics,” J. Math. Kyoto U. 13 (1972), 31–47, was effectively used in Siu–Yau’s proof of Frankel’s Conjecture. It should also be noted that the method of reduction modulo \( p \) in Mori’s proof of Hartshorne’s Conjecture became a clue to the Mori theory on the minimal model program of projective algebraic manifolds.
The Kobayashi–Hitchin correspondence for vector bundles in (6) states that a holomorphic vector bundle \( E \) over a compact Kähler manifold is stable in the sense of Mumford–Takemoto if and only if \( E \) admits a Hermitian–Einstein metric. Kobayashi and Lübke proved the “if” part, while the “only if” part, conjectured by Kobayashi and Hitchin, was proved finally by Donaldson and Uhlenbeck–Yau.
Since 1995 Professor Kobayashi regularly attended our annual workshop on complex geometry at Sugadaira, Japan.
