arXiv subjects

Junjiro Noguchi

Publications and source records attributed to Junjiro Noguchi.

At least 19 recordsLinked to original sources

arXiv2022

On Kiyoshi Oka's Unpublished Papers in 1943

In 1943 from September to December Kiyoshi Oka wrote a series of papers numbered from VII to XI, as the research reports to Teiji Takagi (then, Professor of Tokyo Imperial University), in which he solved affirmatively the so-called Levi Problem (Hartogs' Inverse Problem termed by Oka) for unramified Riemann domains over $\C^n$. This problem which had been left open for more than thirty years then, was the last one of the Three Big Problems summarized by Behnke--Thullen 1934. The papers were hand-written in Japanese, consist of pp.~108 in total, and have not been published by themselves. The aim of the present article is to provide an English translation of the most important, last paper (Part II) with preparation (Part I). At the end of Part I we will discuss a problem which K. Oka left and is still open.

math.CV↗

arXiv2022

Analytic Ax-Schanuel Theorem for semi-abelian varieties and Nevanlinna theory

The purpose of this paper is to explore Nevanlinna theory of the entire curve $\exh_A f:=(\exp_Af,f):\C \to A \times \Lie(A)$ associated with an entire curve $f: \C \to \Lie(A)$, where $\exp_A:\Lie(A)\to A$ is an exponential map of a semi-abelian variety $A$. Firstly we give a Nevanlinna theoretic proof to the {\em analytic Ax-Schanuel Theorem} for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series (Ax-Schanuel Theorem). We assume some non-degeneracy condition for $f$ such that the elements of the vector-valued function $f(z)-f(0) \in \Lie(A)\iso \C^n$ are $\Q$-linearly independent in the case of $A=(\C^*)^n$. Then by making use of the Log Bloch-Ochiai Theorem and a key estimate which we show, we prove that $\td_\C\, \exh_A f \geq n+ 1$. Our next aim is to establish a {\em 2nd Main Theorem} for $\exh_A f$ and its $k$-jet lifts with truncated counting functions at level one.

math.CV↗

arXiv2021

A brief proof of Bochner's tube theorem and a generalized tube

The aim of this note is firstly to give a new brief proof of classical Bochner's Tube Theorem (1938) by making use of K. Oka's Boundary Distance Theorem (1942), showing directly that two points of the envelope of holomorphy of a tube can be connected by a line segment. We then apply the same idea to show that if an unramified domain $\mathfrak{D}:=A_1+iA_2 \to \mathbf{C}^n$ with unramified real domains $A_j \to \mathbf{R}^n$ is pseudoconvex, then the both $A_j$ are univalent and convex (a generalization of Kajiwara's theorem). From the viewpoint of this result we discuss a generalization by M. Abe with giving an example of a finite tube over $\mathbf{C}^n$ for which Abe's theorem no longer holds. The present method may clarify the point where the (affine) convexity comes from.

math.CV↗

arXiv2020

Analytic and rational sections of relative semi-abelian varieties

The hyperbolicity statements for subvarieties and complements of hypersurfaces in abelian varieties admit arithmetic analogues, due to Faltings (and Vojta for the semi-abelian case). In Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018) by the second author, an analogy between the analytic and arithmetic theories was shown to hold also at proof level, namely in a proof of Raynaud's theorem (Manin-Mumford Conjecture). The first aim of this paper is to extend to the relative setting the above mentioned hyperbolicity results. We shall be concerned with analytic sections of a relative (semi-)abelian scheme over an affine algebraic curve. These sections form a group; while the group of rational sections (the Mordell-Weil group) has been widely studied, little investigation has been pursued so far on the group of the analytic sections. We take the opportunity of developing some basic structure of this apparently new theory, defining a notion of height or order functions for the analytic sections, by means of Nevanlinna theory.

math.CV↗

arXiv2018

A Weak Coherence Theorem and Remarks to the Oka Theory

The proofs of K. Oka's Coherence Theorems are based on Weierstrass' Preparation (division) Theorem. Here we formulate and prove a Weak Coherence Theorem without using Weierstrass' Preparation Theorem, but only with power series expansions: The proof is almost of linear algebra. Nevertheless, this simple Weak Coherence Theorem suffices to give other proofs of the Approximation, Cousin I/II, and Levi's (Hartogs' Inverse) Problems even in simpler ways than those known, as far as the domains are non-singular; they constitute the main basic part of the theory of several complex variables. The new approach enables us to complete the proofs of those problems in quite an elementary way without Weierstrass' Preparation Theorem or the cohomology theory of Cartan--Serre, nor L2-dbar method of Hoermander.

math.CV↗

arXiv2018

A Brief Chronicle of the Levi (Hartogs' Inverse) Problem, Coherence and an Open Problem

Here we chronologically summarize briefly the developments of the Levi (Hartogs' Inverse) Problem together with the notion of coherence and its solution, shedding light on some records which have not been discussed in the past references. In particular, we will discuss K. Oka's unpublished papers 1943 which solved the Levi (Hartogs' Inverse) Problem for unramified Riemann domains of arbitrary dimension $n \geq 2$, usually referred as it was solved by Oka IX in 1953, H.J. Bremermann and F. Norguet in 1954 for univalent domains, independently. At the end we emphasize an open problem in a ramified case.

math.CV↗

arXiv2017

An application of the value distribution theory for semi-abelian varieties to problems of Ax-Lindemann and Manin-Mumford types

The aim of this paper is to prove a theorem of Ax-Lindemann type for complex semi-abelian varieties as an application of a big Picard theorem proved by the author in 1981, and then apply it to prove a theorem of classical Manin-Mumford Conjecture for semi-abelian varieties, which was proved by M. Raynaud 1983, M. Hindry 1988, ....., and Pila-Zannier 2008 by a different method from others, which is most relevant to our's. The present result might be a first instance of a direct connection between the value distribution theory of holomorphic maps and the arithmetic (Diophantine) theory over algebraic number fields, while there have been many analogies between them.

math.NT↗

arXiv2015

A Scalar Associated with the Inverse of Some Abelian Integrals and a Ramified Riemann Domain

We introduce a positive scalar function $ρ(a, Ω)$ for a domain $Ω$ of a complex manifold $X$ with a global holomorphic frame of the cotangent bundle by closed Abelian differentials, which heuristically measure the distance from $a \in Ω$ to the boundary $\delΩ$. We prove an {\em estimate of Cartan--Thullen type with $ρ(a, Ω)$} for holomorphically convex hulls of compact subsets. In one dimensional case, we apply the obtained estimate of $ρ(a, Ω)$ to give a new proof of Behnke-Stein's Theorem for the Steiness of open Riemann surfaces. We then use the same idea to deal with the Levi problem for ramified Riemann domains over $\C^n$. We obtain some geometric conditions in terms of $ρ(a, X)$ which imply the validity of the Levi problem for a finitely sheeted Riemann domain over $\C^n$.

math.CV↗

arXiv2011

On Oka's Extra-Zero Problem

After the solution of Cousin II problem by K. Oka III in 1939, he thought an {\it extra-zero problem} in 1945 (his posthumous paper) asking if it is possible to solve an arbitrarily given Cousin II problem adding some extra-zeros whose support is disjoint from the given one. By the secondly named author, some special case was affirmatively confirmed in dimension two and a counter-example in dimension three or more was given. The purpose of the present paper is to give a complete solution of this problem with examples and to discuss some new questions.

math.CV↗

arXiv2011

Another Direct Proof of Oka's Theorem (Oka IX)

In 1953 K. Oka IX solved in first and in a final form Levi's problem (Hartogs' inverse problem) for domains or Riemann domains over $\C^n$ of arbitrary dimension. Later on a number of the proofs were given; cf.\ e.g., Docquier-Grauert's paper in 1960, R. Narasimhan's paper in 1961/62, Gunning-Rossi's book, and Hörmander's book (in which the holomorphic separability is pre-assumed in the definition of Riemann domains and thus the assumption is stronger than the one in the present paper). Here we will give another direct elementary proof of Oka's Theorem, relying only on Grauert's finiteness theorem by the {\it induction on the dimension} and the {\it jets over Riemann domains}; hopefully, the proof is most comprehensive.

math.CV↗

arXiv2011

Connections and the Second Main Theorem for Holomorphic Curves

By means of $C^\infty$-connections we will prove a general second main theorem and some special ones for holomorphic curves. The method gives a geometric proof of H. Cartan's second main theorem in 1933. By applying the same method, we will prove some second main theorems in the case of the product space $(\pone)^2$ of the Riemann sphere.

math.CV↗

arXiv2011

Order of Meromorphic Maps and Rationality of the Image Space

Let $ι: \C^2 \hookrightarrow S$ be a compactification of the two dimensional complex space $\C^2$. By making use of Nevanlinna theoretic methods and the classification of compact complex surfaces K. Kodaira proved in 1971 (\cite{ko71}) that $S$ is a rational surface. Here we deal with a more general meromorphic map $f: \C^n \to X$ into a compact complex manifold $X$ of dimension $n$, whose differential $df$ has generically rank $n$. Let $ρ_f$ denote the order of $f$. We will prove that if $ρ_f<2$, then every global symmetric holomorphic tensor must vanish; in particular, {\it if $\dim X=2$ and $X$ is kähler, then $X$ is a rational surface. Without the kähler condition there is no such conclusion, as we will show by a counter-example using a Hopf surface.} This may be the first instance that the kähler or non-kähler condition makes a difference in the value distribution theory.

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arXiv2009

A New Unicity Theorem and Erdos' Problem for Polarized Semi-Abelian Varieties

In 1988 P. Erdös asked if the prime divisors of $x^n -1$ for all $n=1,2, >...$ determine the given integer $x$; the problem was affirmatively answered by Corrales-Rodorigáñez and R. Schoof in 1997 together with its elliptic version. Analogously, K. Yamanoi proved in 2004 that the support of the pull-backed divisor $f^{*}D$ of an ample divisor on an abelian variety $A$ by an algebraically non-degenerate entire holomorphic curve $f: \C \to A$ essentially determines the pair $(A, D)$. By making use of a recent theorem of Noguchi-Winkelmann-Yamanoi in Nevanlinna theory, we here deal with this problem for semi-abelian varieties: namely, given two polarized semi-abelian varieties $(A_1, D_1)$, $(A_2,D_2)$ and entire non-degenerate holomorphic curves $f_i:\C\to A_i$, $i=1,2$, we classify the cases when the inclusion $\supp f_1^*D_1\subset \supp f_2^* D_2$ holds. We also apply a result of Corvaja-Zannier on linear recurrence sequences to prove an arithmetic counterpart.

math.CV↗

arXiv2005

The Second Main Theorem for Holomorphic Curves into Semi-Abelian Varieties II

We establish the second main theorem with the best truncation level one for an entire holomorphic curve $f:\C \to A$ into a semi-abelian variety $A$ and an arbitrary effective reduced divisor $D$ on $A$; the low truncation level is important for applications. We will actually prove this for the jet lifts of $f$. Finally we give some applications, including the solution of a problem posed by Mark Green.

math.CV↗

arXiv2005

Degeneracy of Holomorphic Curves into Algebraic Varieties

Applying the Second Main Theorem we deal with the algebraic degeneracy of entire holomorphic curves from the complex plane into a complex algebraic normal variety of positive log Kodaira dimension that admits a finite proper morphism to a semi-abelian variety. We will also discuss applications to the Kobayashi hyperbolicity problem.

math.CV↗

arXiv2002

Bounds for Curves in Abelian Varieties

A uniform bound of intersection multiplicities of curves and divisors on abelian varieties is proved by algebraic geometric methods. It extends and improves a result obtained by A. Buium with a different method based on Kolchin's differential algebra. The problem is modeled after the abc-Conjecture of Masser-Oesterle for abelian varieties over the function field of a curve. As an application a finiteness theorem is proved for maps from a curve into an abelian variety omitting an ample divisor.

math.AG↗

arXiv2002

A note on Jets of Entire Curves in Semi-Abelian Varieties

We prove a product decomposition of the Zariski closure of the jet lifts of a holomorphic map f from C into a semi-abelian variety A, provided that f is of finite order. On the other hand, by giving an example of such a map f into a three dimensional abelian variety we show that this product decomposition does not hold in general; there was a gap in the proofs of some articles which claimed that such a product decomposition is always true.

math.AG↗

arXiv1999

The Second Main Theorem for Holomorphic Curves into Semi-Abelian Varieties

Let f:C -> A be an entire holomorphic curve into a semi-Abelian variety A. Then the Zariski closure of f(C) is a translate of a semi-Abelian subvariety of A (logarithmic Bloch-Ochiai's theorem). The purpose of the present paper is to establish a quantitative version of the above result for such f i.e., the second main theorem and the defect relation.

math.CV↗