Seiki Mori

Defects of holomorphic curves Into P n (C) for rational moving targets and a space of meromorphic mappings

In this paper, we show an elimination theorem of defects of a holomorphic curve into P n (C) for rational moving targets, and also we show that the set of meromorphic mappings without defects is dense in the space of all meromorphic mappings into P n (C).

Elimination of Defects of Meromorphic Mappings by Small Deformation

For a nondegenerate meromorphic mapping f of C m into the complex projective space P n (C),Nevanlinna’s defect relation asserts that the set of hyperplanes with a positive Nevanlinna deficiency in a set X ⊂ P n (C)* of hyperplanes in general position is at most countable. (Such a hyperplane is called a deficient hyperplane or a defect.) Further, Sadullaev [6] proved that the set of hyperplanes with a positive Valiron deficiency is of capacity zero (or it is a locally pluripolar set), and also we observe that the set of Valiron deficient hyperplanes has projective logarithmic capacity zero in the sense of Molzon-Shiffman-Sibony [3]. These results assert that defects of a meromorphic mapping are very few, and sometimes Nevanlinna theory is called the equidistribution theory.

On one new singular direction of meromorphic functions

In this article, we deal with one new singular direction of meromorphic function of infinite order concerning shared-values. Moreover, the example shows the condition is necessary.

Topics on meromorphic mappings and defects

This note is a survey report based on the talk at the International Workshop on Value Distribution Theory and its Applications, at Weihai, China.

Deficiencies of Meromorphic Functions

Suppose that f is a meromorphic function and g is an entire function. In this paper, we study the relations between the deficiency of f (g) and the one of f, g.

A Space of Meromorphic Mappings and Defects of a Meromorphic Mapping into Pn(C)

Nevanlinna defect relations ware established for various cases, for example, holomorphic (or meromorphic) mappings of C m into a complex projective space P n (C) for constant or moving targets of hyperplanes (arbitrary m ≥ 1 and n ≥ 1), or holomorphic mappings of an affine variety A of dimension m into a projective algebraic variety V of dimension n for divisors on V (m ≥ n ≥ 1), and so on. On the other hand, the size of a set of (Valiron) deficient hyperplanes or deficient di visors are investigated. (e.g. Mori[1], Sadullaev [4].) Nevanlinna theory asserts that for each holomorphic (or meromorphic) mapping, Nevanlinna defects or Valiron defects of a mapping are very few. Recently the author [1], [2] and [3] proved that for a transcendental meromorphic mapping (or holomorphic curve) f of C m (or m= 1) into P n (C), we can eliminate all deficient hyperplanes, hypersurfaces of degree at most a given integer d or rational moving targets in P n (C) by a small deformation of the mapping. Here a samll deformation \(\tilde f\) of f means that their order functions T f (r) and \( {{T}_{{\tilde{f}}}}(r) \) satisfy \(\left| {{T_f}(r) - {T_{\tilde f}}(r)} \right| \leqslant o({T_f}(r))\), as r tends to infinity. Therefore,it seems to me that mappings with positive deficiencies are very few in a space of meromorphic mappings into P n (C).

Elimination of Defects of Non-Archchimedean Holomorphic Curves

In this paper, we will prove that for any transcendental holomorphic curve f :k → ℙ n (k) where k is an algebraically closed field of characteristic zero, complete for a non-trivial non-Archimedean absolute value, we can eliminate all Nevanlinna defects by small deformations if there exists at least a hyperplane whose Valiron deficiency for f is zero.

Some properties on the growth of the nevanlinna proximity function and the projective logarithmic capacity

It is known that for an entire function f of into the set of points a with limr-x mf(r,a)=∞ is of logarithmic capacity zero, and conversely given any set E of logarithmic capacity zero in there exi...