A space of meromorphic mappings and an elimination of defects (Q5950090)

This is a summary report of the author's recent articles. NEWLINENEWLINENEWLINELet \(f:\mathbb{C}^m\rightarrow\mathbb{P}^n(\mathbb{C})\) be a transcendental meromorphic mapping. The author [Ann. Acad. Sci. Fenn. Math., 24, No. 1, 89-104 (1999; Zbl 0930.32011)] proved that there is a small deformation of \(f\) which has no Nevanlinna deficient hyperplanes in \(\mathbb{P}^n(\mathbb{C})\), where a ``small deformation'' of \(f\) means a meromorphic mapping \(g:\mathbb{C}^m\rightarrow\mathbb{P}^n(\mathbb{C})\) such that their order functions \(T_f(r)\) and \(T_g(r)\) satisfy \(T_g(r)=T_f(r)+o(T_f(r))\). For the case \(m=1\), he also proved that there is a small deformation of \(f\) which has no Nevanlinna deficient hypersurfaces of degree \(\leq d\) for each given positive integer \(d\) [Recent Developments in Complex Analysis and Computer Algebra, Kluwer Acad. Publ., Int. Soc. Anal. Appl. Comput. 4, 247-258 (1999; Zbl 0961.32018)], or deficient rational moving targets [Complex Variables 43, No. 3-4, 363-379 (2000; Zbl 1022.30032)]. Furthermore, the author shows that the mappings without Nevanlinna defects are dense in a space of transcendental meromorphic mappings.