Katsutoshi Yamanoi

The second main theorem for small functions and related problems

We shall establish the following three results in more general forms. (1) The second main theorem for small functions. Let f be a meromorphic function on the complex plane C. Let a1, . . . , aq be distinct meromorphic functions on C. Assume that ai are small with respect to f ; i.e., T (r, a) 0 (Corollary 1). Here as usual in Nevanlinna theory, the terms T (r, f) and N(r, ai, f) denote for the characteristic function and the truncated counting function, respectively. (2) Application to functional equations. Let KC be the field of meromorphic functions on C. For a function ψ : R>0 → R, put KψC = {a ∈ KC; T (r, a) ≤ O(ψ(r)) ||}, which is a subfield of KC. Then the following holds: Let F (x, y) ∈ KψC [x, y] be a polynomial in two variables over K ψ C . Assume that the curve F (x, y) = 0 over KψC has genus greater than one. If ζ1, ζ2 ∈ KC satisfy the functional equation F (ζ1, ζ2) = 0, then both ζ1 and ζ2 are contained in KψC (Corollary 2). (3) Height inequality for curves over function fields. Let k be a function field of one variable over C. Let X be a smooth projective curve over k, let D ⊂ X be a reduced divisor, let L be an ample line bundle on X and let e > 0. Then we have hk,KX (D)(P ) ≤ N (1) k,S(D,P ) + dk(P ) + ehk,L(P ) +Oe(1) for all P ∈ X(k)\D (Theorem 5). Here the notations are introduced in [V1], [V3] (see also section 9). Our proof uses Ahlfors’ theory of covering surfaces and the geometry of the moduli space of q-pointed stable curves of genus 0.

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.

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.

Defect relation for rational functions as targets

Abstract The second main theorem in Nevanlinna theory is proved when targets are rational functions. We use Ahlfors’ theory of covering surfaces for a proof.

ON THE TRUNCATED SMALL FUNCTION THEOREM IN NEVANLINNA THEORY

We prove a second main theorem type estimate in Nevanlinna theory when a target space is a family of curves. This estimate unifies the truncated q-small function theorem, and the height inequality for curves over function fields.

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.

Holomorphic curves in algebraic varieties of maximal albanese dimension

We prove a second main theorem type estimate in Nevanlinna theory for holomorphic curves f : Y → X from finite covering spaces Y → ℂ of the complex plane ℂ into complex projective manifolds X of maximal albanese dimension. If X is moreover of general type, then this implies that the special set of X is a proper subset of X. For a projective curve C in such X, our estimate also yields an upper bound of the ratio of the degree of C to the geometric genus of C, provided that C is not contained in a proper exceptional subset in X.

A generalization of a completeness lemma in minimal surface theory

We settle a question posed by Umehara and Yamada, which generalizes a completeness lemma useful in differential geometry.