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## Über dieses Buch

This volume is dedicated to the memory of Shoshichi Kobayashi, and gathers contributions from distinguished researchers working on topics close to his research areas. The book is organized into three parts, with the first part presenting an overview of Professor Shoshichi Kobayashi’s career. This is followed by two expository course lectures (the second part) on recent topics in extremal Kähler metrics and value distribution theory, which will be helpful for graduate students in mathematics interested in new topics in complex geometry and complex analysis. Lastly, the third part of the volume collects authoritative research papers on differential geometry and complex analysis.

Professor Shoshichi Kobayashi was a recognized international leader in the areas of differential and complex geometry. He contributed crucial ideas that are still considered fundamental in these fields. The book will be of interest to researchers in the fields of differential geometry, complex geometry, and several complex variables geometry, as well as to graduate students in mathematics.

## Inhaltsverzeichnis

### In Memory of Professor Shoshichi Kobayashi

As I stop and reflect on the life and academic achievements of Professor Shoshichi Kobayashi, “the beautiful theorem” on how a great mathematician should be comes to mind. His original and sharp theorems read like masterpiece short stories. His writings splendidly harmonize and play like a symphony. Though I am aware of my inability to reach the height of his talent, I dare to write this article to introduce the fine personal character and remarkable academic achievements of Professor Shoshichi Kobayashi.
Takushiro Ochiai

### Events Surrounding the Birth of the Kobayashi Metric

I first met Sho in 1962 in one of the AMS Summer Institutes in Santa Barbara. I had just finished my first year as a graduate student at MIT and he told me he was on his way to Berkeley. We ended up being colleagues for 47 years when I myself got to Berkeley in 1965. Although as colleagues we could not help but run into each other often, I think it was in the ten or so years from 1980 to 1990 that I had extended contact with him every week, when he drove me home after each differential geometry seminar late in the day on Friday.We had to walk a bit before we could get to his car and that gave us even more of a chance to chat and gossip. I am afraid the intellectual quality of the conversations was not particularly high, but the entertainment value was off the charts. It was most enjoyable. However, the one thing that has stuck in my mind about Sho all these years is probably the fortuitous confluence of events surrounding the discovery of the Kobayashi metric in 1966.
Hung-Hsi Wu

### Academic Genealogy of Shoshichi Kobayashi and Individuals Who Influenced Him

Professor Yoshiaki Maeda, a co-editor of this volume, kindly asked me to prepare an article on academic genealogy of my brother Shoshichi Kobayashi. In the olden times, it would have been a formidable task for any individual who is not in the same field as the subject mathematician to undertake a genealogy search. Luckily, the “Mathematics Genealogy Project” (administered by the Department of Mathematics, North Dakota State University) [1] and Wikipedia [2] which document almost all notable mathematicians, provided the necessary information for me to write this article. Before I undertook this study, I knew very little about Shoshichi’s academic genealogy, except for his advisor Kentaro Yano at the University of Tokyo and his Ph.D. thesis advisor Carl Allendoerfer at the University of Washington, Seattle.
Hisashi Kobayashi

### Algebraic Differential Equations for Entire Holomorphic Curves in Projective Hypersurfaces of General Type: Optimal Lower Degree Bound

Let $$X\;=\;X^{n}\;\subset\;\mathbb{P}^{n+1}\left(\mathbb{C}\right)$$ be a geometrically smooth projective algebraic complex hypersurface. Using Green–Griffiths jets, we establish the existence of nonzero global algebraic differential equations that must be satisfied by every nonconstant entire holomorphic curve $$\mathbb{C}\;\rightarrow\;X$$ if X is of general type, namely if its degree d satisfies the optimal possible lower bound
$$d\;\geqslant\;n+3.$$
Joël Merker

### Kobayashi Hyperbolicity and Lang’s Conjecture

The purpose of the present article is to survey the development of the theory of Mordell’s Conjecture and Lang’s Conjecture in relation with the Kobayashi hyperbolicity, and their analogues over function fields. We will also discuss the deeply related Shafarevich Conjecture and the related results. In last, we will mention some open problems.
Junjiro Noguchi

### A Lemma on Hartogs Function and Application to Levi Flat Hypersurfaces in Hopf Surfaces

The Levi form of the Hartogs function is computed for the domains with Levi flat boundary. The result is applied to the classification of Levi flat hypersurfaces in Hopf surfaces.
Takeo Ohsawa

### On the Extremal Measure on a Complex Manifold

We study the extremal measure on a complex manifold introduced in [T4] and prove the logarithmic plurisubhamonic variation property under certain mild conditions. We also define a dynamical system of extremal measures and prove that it converges to the Kähler–Einstein volume form as in [T3], if the manifold is canonically polarized (Theorem 4.1).
Hajime Tsuji

### A Lang Exceptional Set for Integral Points

In 1986 and 1991, Serge Lang defined holomorphic, diophantine, and geometric exceptional sets of a complete variety over C, over a number field, or over a field of characteristic zero, respectively, and conjectured that they should coincide when defined.
This talk examined the possibility of extending this definition to holomorphic curves or integral points in quasi-projective varieties. A central question that arises is, given an abelian (or semiabelian) variety A and a Zariskiclosed subset Z of codimension ≥ 2, can one find a nonconstant holomorphic curve in A \ Z with Zariski-dense image, or a Zariski-dense set of integral points on A \ Z? This paper proves this result for holomorphic curves (this is quite easy). For integral points, however, the question remains open. Some partial results are obtained.
Paul Vojta

### Kobayashi Hyperbolicity and Higher-dimensional Nevanlinna Theory

This note is a survey concerning Kobayashi hyperbolicity problem and higher dimensional Nevanlinna theory. The central topic of this note is a famous open problem to characterize which projective varieties are Kobayashi hyperbolic. We shall review some recent progress on this problem and explain some technical details of the role of Nevanlinna theory in this problem.
Katsutoshi Yamanoi

### Geometry and Arithmetic on the Siegel–Jacobi Space

The Siegel–Jacobi space is a non–symmetric homogeneous space which is very important geometrically and arithmetically. In this paper, we discuss the theory of the geometry and the arithmetic of the Siegel–Jacobi space.
Jae-Hyun Yang

### On the Pseudonorm Project of Birational Classification of Algebraic Varieties

The birational classification of complex varieties of general type has been one of the most crucial and challenging task in the development of algebraic geometry, since the time Kodaira accomplished his monumental work on the classification of compact complex surfaces. The current note serves as an exposition of a project towards birational classification of complex varieties of general type, which I initiated in 2008. The central role of this project is played by pseudonorms, some analytically defined norm-like functions, on the pluricanonical spaces of compact complex manifolds. We also survey on different approaches towards the birational Torelli type theorem, which is the initial step of the project, and indicate some directions for future developments.
Shing-Tung Yau

### The Weighted Laplacians on Real and Complex Metric Measure Spaces

In this short note we compare the weighted Laplacians on real and complex (Kähler) metric measure spaces. In the compact case Kähler metric measure spaces are considered on Fano manifolds for the study of Kähler–Einstein metrics while real metric measure spaces are considered with Bakry–Émery Ricci tensor. There are twisted Laplacians which are useful in both cases but look alike each other. We see that if we consider noncompact complete manifolds significant differences appear.
Akito Futaki

### Locally Conformally Kähler Structures on Homogeneous Spaces

We will discuss in this paper homogeneous locally conformally Kähler (or shortly homogeneous l.c.K.) manifolds and locally homogeneous l.c.K. manifolds from various aspects of study in the field of l.c.K. geometry. We will provide a survey of known results along with some new results and observations; in particular we make a complete classification of 4-dimensional homogeneous and locally homogeneous l.c.K. manifolds in terms of Lie algebras.
Keizo Hasegawa, Yoshinobu Kamishima

### A Note on Vanishing Theorems

On a Riemannian manifold we define a one-parameter family of Laplacians acting on sections of any bundle associated to the principal frame bundle via a representation, and show how various examples fit into this framework.
Nigel Hitchin

### Dupin Hypersurfaces in Lie Sphere Geometry

We present a moving frames proof, with motivation and context, that all nonumbilic Dupin immersions of a surface are Lie sphere congruent to each other.
Gary R. Jensen

### The Donaldson–Futaki Invariant for Sequences of Test Configurations

In this paper, given a polarized algebraic manifold (X,L), we define the Donaldson–Futaki invariant $$F_1\left(\{\mu_{i}\}\right)$$ for a sequence $$\{\mu_{i}\}$$ of test configurations for (X,L) of exponents lisatisfying
$$l_i\rightarrow\;\infty,\quad \mathrm{as} \ j\rightarrow\;\infty.$$
This then allows us to define a strong version of K-stability or K-semistability for (X,L). In particular, (X,L) will be shown to be K-semistable in this strong sense if the polarization class $$c_1\left(L\right)_\mathbb{R}$$ admits a constant scalar curvature Kähler metric.
Toshiki Mabuchi

### Strong K-stability and Asymptotic Chow-stability

For a polarized algebraic manifold (X,L), let T be an algebraic torus in the group Aut(X) of all holomorphic automorphisms of X. Then strong relative K-stability (cf. [6]) will be shown to imply asymptotic relative Chow-stability. In particular, by taking T to be trivial, we see that asymptotic Chow-stability follows from strong K-stability.
Toshiki Mabuchi, Yasufumi Nitta

### Traces and Characteristic Classes in Infinite Dimensions

We survey geometric constructions of characteristic classes associated to certain infinite rank bundles on the loop space LM of a manifold M. There are two types of classes, which arise from applying either the leading order trace or the Wodzicki residue to the curvature of natural connections on TLM, as the curvature forms take values in pseudodifferential operators. The leading order classes lead to a restatement of the S 1 -index theorem on LM, provide generators for the cohomology of loop groups, and for Maps(S 2, M) are related to Gromov-Witten invariants. The Wodzicki classes have applications to the topology of diffeomorphism groups of certain circle bundles over Kaehler surfaces.
Yoshiaki Maeda, Steven Rosenberg

### Moment Map Description of the Cartan–Münzner Polynomials of Degree Four

This is a survey of the description of all the known Cartan–Münzner polynomials of degree four in terms of the moment map of certain group actions.
Reiko Miyaoka

### Ribaucour Pairs Corresponding to Dual Pairs of Conformally Flat Hypersurfaces

We discuss generic conformally flat hypersurfaces of dimension 3 in Euclidean space and, in particular, the duality of such hypersurfaces. There exists a one-to-one correspondence between dual pairs of conformally flat hypersurfaces and Ribaucour pairs of their canonical Guichard nets. These Ribaucour pairs encode important geometric information for dual pairs of hypersurfaces: the Ribaucour pairs determine the geometry of dual pairs of hypersurfaces intrinsically and also extrinsically. In this paper, a method to construct Ribaucour pairs of canonical Guichard nets directly from one Guichard net is proposed.
Udo Hertrich-Jeromin, Yoshihiko Suyama

### Geometry of Symmetric R-spaces

We look back at the history of symmetric R-spaces and give a survey of the geometry of symmetric R-spaces including the author’s recent results.
Makiko Sumi Tanaka
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