Celebratio Mathematica

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:

1. the Kobay­ashi in­trins­ic pseudo-dis­tance,
2. the Kobay­ashi hy­per­bol­i­city and meas­ure hy­per­bol­i­city,
3. pro­ject­ively in­vari­ant dis­tances for af­fine and pro­ject­ive dis­tances,
4. the study of com­pact com­plex man­i­folds with pos­it­ive Ricci curvature,
5. filtered Lie al­geb­ras and geo­met­ric struc­tures, and
6. 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.