Junjiro Noguchi

Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties

Nevanlinnas lemma on logarithmic derivatives played an essential role in the proof of the second main theorem for meromorphic functions on the complex plane C (c£, e.g., [17]). In [19, Lemma 2.3] it was generalized for entire holomorphic curves /: C->M in a compact complex manifold M (Lemma 2.3 in [19] is still valid for non-Kahler M). Here we call, in general, a holomorphic mapping from a domain of C or a Riemann surface into M a holomorphic curve in M, and sometimes use it in the sense of its image if no confusion occurs. Applying the above generalized lemma on logarithmic derivatives to holomorphic curves /: C -> V in a complex projective algebraic smooth variety V and making use of Ochiai [22, Theorem A], we had an inequality of the second main theorem type for / and divisors on V (see [19, Main Theorem] and [20]). Other generalizations of Nevanlinnas lemma on logarithmic derivatives were obtained by Nevanlinna [16], Griffiths-King [10, § 9] and Vitter [23]. In this paper we first deal with holomorphic curves /: J* -* M from the punctured disc J* = {\z\ >̂ 1} with center at the infinity oo of the Riemann sphere into a compact Kahler manifold M. Our first aim is to prove the following lemma on logarithmic derivatives which is a generalization of Nevanlinna [16, IΠ, p. 370] and will play a crucial role in §§ 3 and 4 (see § 1 as to the notation):

The second main theorem for holomorphic curves into semi-Abelian varieties

We establish the second main theorem with the best truncation level one Tf (r;L(D)) N1(r; f∗D) + Tf (r)|| for an entire holomorphic curve f : C → 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.

Nevanlinna theory in several complex variables and Diophantine approximation

Nevanlinna Theory of Meromorphic Functions.- First Main Theorem.- Differentiably Non-Degenerate Meromorphic Maps.- Entire Curves into Algebraic Varieties.- Semi-Abelian Varieties.- Entire Curves into Semi-Abelian Varieties.- Kobayashi Hyperbolicity.- Nevanlinna Theory over Function Fields.- Diophantine Approximation.- Bibliography.- Index.- Symbols

On holomorphic curves in semi-Abelian varieties

The algebraic degeneracy of holomorphic curves in a semi-Abelian variety omitting a divisor is proved (Langs conjecture generalized to semi-Abelian varieties) by making use of the {\it jet-projection method} and the logarithmic Wronskian jet differential after Siu-Yeung. We also prove a structure theorem for the locus which contains all possible image of non-constant entire holomorphic curves in a semi-Abelian variety omitting a divisor.

Logarithmic Jet Spaces and Extensions of de Franchis’ Theorem

Jet bundles and logarithmic 1-forms play an important role in the study of the value distribution of meromorphic mappings into algebraic varieties (cf. [01] [G-G1] and [N1∿41). On the other hand, logarithmic vector fields as well as logarithmic 1-forms have been used in the study of Gauss-Manin connection and singularities (cf. [S1]).

The second main theorem for holomorphic curves into semiabelian varieties II

Abstract We establish the second main theorem with the best truncation level one for the k-jet lift Jk (ƒ) : → Jk (A) of an algebraically non-degenerate entire holomorphic curve ƒ : → A into a semi-abelian variety A and an arbitrary algebraic reduced subvariety Z of Jk (A); the low truncation level is important for applications. Finally we give some applications, including the solution of a problem posed by Mark Green (1974). 2000 Mathematics Subject Classification: 32H30.

The value distribution of holomorphic curves into semi-Abelian varieties

Abstract 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–Ochiais theorem). We establish a quantitative version of the above result for such f , i.e., the second main theorem and the defect relation.

A note on jets of entire curves in semi-abelian varieties

We prove a product decomposition of the Zariski closure of every jet lift of an entire curve f:C→A into a semi-abelian variety A, provided that f is of finite order. On the other hand, by giving an example of 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 [2], Proposition 1.8 (ii) and of [6], Theorem 2.2.

Some results in view of Nevanlinna theory

Here we discuss and survey some results on rational points of algebraic varieties and Nevanlinna theory in relation to Lang’s conjectures and Vojta’s. We will also announce new results in the case of function fields and also the second main theorem for holomorphic curves into semiabelian varieties.

Oka’s First Coherence Theorem

We study the local properties of holomorphic functions. The main object is Oka’s Coherence Theorem that plays the most fundamental and important role in analytic function theory in several variables.