A new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variables

Abstract

Let c ∈ C m , f : C m → P n (C) be a linearly nondegenerate meromorphic mapping over the field P c of c-periodic meromorphic functions in C m , and let H j (1 ≤ j ≤ q) be q(> 2N − n + 1) hyperplanes in N-subgeneral position of P n (C). We prove a new version of the second main theorem for meromorphic mappings of hyperorder strictly less than one without truncated multiplicity by considering the Casorati determinant of f instead of its Wronskian determinant. As its applications, we obtain a defect relation, a uniqueness theorem and a difference analogue of generalized Picard theorem.

The second main theorem for holomorphic curves intersecting hypersurfaces with Casorati determinant into complex projective spaces

“…We note that the main result can be also extended to the case for meromorphic mappings from C m into complex projective spaces P n (C) by the standard process of averaging over the complex lines in the complex space C m , we refer to see two related references [2,3] on this topic. Remark 1.1.…”

Logarithmic difference lemma in several complex variables and partial difference equations

“…under the assumption of (2). We will also show the explicit expression of o(T (r, f )) in the logarithmic difference lemma for the special case whenever f is of finite order as for any ε(> 0).…”

“…(ii). By the new version of the logarithmic difference lemma, all the second main theorem and Picard type theorem for meromorphic mappings from C m into complex projective spaces P n (C) obtained in [24,1,2] (including also [16,36,25,3]) can be improved under the assumption of (5).…”

“…In 2012, Korhonen [24] firstly obtained the difference version of logarithmic derivative lemma (shortly, we may say logarithmic difference lemma) for meromorphic functions on C m with hyperorder strictly less than 2 3 , and then used it to consider a class of partial difference equations in the same paper. In [2], We assume that the readers are familiar with the basic notations and results on Nevanlinna theory for meromorphic functions in several complex variables (refer to see, for examples [14,33,32]).…”

“…Let c ∈ C m \{0}. Motivated by the ideas of [4,39], we will prove an improvement of the logarithmic difference lemma in several complex variables [24,2] that (which implies that the hyperorder is rather than just strictly less than one). Then from it, we get the relation…”