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

Abstract

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).