In this paper, we establish a second main theorem for holomorphic maps with finite growth index on complex discs intersecting arbitrary families of hypersurfaces (fixed and moving) in projective varieties, which gives an above bound of the sum of truncated defects. Our result also generalizes and improves many previous second main theorems for holomorphic maps from ${\mathbb{C}}$ intersecting hypersurfaces (moving and fixed) in projective varieties.

We investigate Carlson–Griffiths’ equidistribution theory of meormorphic mappings from a complete Kähler manifold into a complex projective algebraic manifold. By using a technique of Brownian motions developed by Atsuji, we obtain a second main theorem in Nevanlinna theory provided that the source manifold is of nonpositive sectional curvature. In particular, a defect relation follows if some growth condition is imposed.

We define and study jet bundles in the geometric orbifold category. We show that the usual arguments from the compact and the logarithmic settings do not all extend to this more general framework. This is illustrated by simple examples of orbifold pairs of general type that do not admit any global jet differential, even if some of these examples satisfy the Green–Griffiths–Lang conjecture. This contrasts with an important result of Demailly (Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, Pure Appl. Math. Q. 7 (2011), 1165–1207) proving that compact varieties of general type always admit jet differentials. We illustrate the usefulness of the study of orbifold jets by establishing the hyperbolicity of some orbifold surfaces, that cannot be derived from the current techniques in Nevanlinna theory. We also conjecture that Demailly's theorem should hold for orbifold pairs with smooth boundary divisors under a certain natural multiplicity condition, and provide some evidence towards it.

Let $\{\mathbf{F}(n)\}_{n\in \mathbb{N}}$ and $\{\mathbf{G}(n)\}_{n\in \mathbb{N}}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set ${\mathcal{N}}$ of natural numbers such that their ratio $\mathbf{F}(n)/\mathbf{G}(n)$ is an integer. In this paper we study an analogue of such a divisibility problem in the complex situation. Namely, we are concerned with the divisibility problem (in the sense of complex entire functions) for two sequences $F(n)=a_{0}+a_{1}f_{1}^{n}+\cdots +a_{l}f_{l}^{n}$ and $G(n)=b_{0}+b_{1}g_{1}^{n}+\cdots +b_{m}g_{m}^{n}$, where the $f_{i}$ and $g_{j}$ are nonconstant entire functions and the $a_{i}$ and $b_{j}$ are non-zero constants except that $a_{0}$ can be zero. We will show that the set ${\mathcal{N}}$ of natural numbers such that $F(n)/G(n)$ is an entire function is finite under the assumption that $f_{1}^{i_{1}}\cdots f_{l}^{i_{l}}g_{1}^{j_{1}}\cdots g_{m}^{j_{m}}$ is not constant for any non-trivial index set $(i_{1},\ldots ,i_{l},j_{1},\ldots ,j_{m})\in \mathbb{Z}^{l+m}$.

It is a well-known result that if a nonconstant meromorphic function $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f$ on $\mathbb{C}$ and its $l$th derivative $f^{(l)}$ have no zeros for some $l\geq 2$, then $f$ is of the form $f(z)=\exp (Az+B)$ or $f(z)=(Az+B)^{-n}$ for some constants $A$, $B$. We extend this result to meromorphic functions of several variables, by first extending the classic Tumura–Clunie theorem for meromorphic functions of one complex variable to that of meromorphic functions of several complex variables using Nevanlinna theory.

We prove a big Picard type extension theoremfor holomorphic maps $f\,:\,X\,-\,A\,\to \,M$, where $X$ is a complex manifold, $A$ is an analytic subvariety of $X$, and $M$ is the complement of the union of a set of hyperplanes in ${{\mathbb{P}}^{n}}$ but is not necessarily hyperbolically imbedded in ${{\mathbb{P}}^{n}}$.

We propose a new approach to the value distribution theory of entire holomorphic curves. We define packing density of Brody curves, and show that it has various non-trivial properties. The packing density of Brody curves can be considered as an infinite dimensional version of characteristic number, and it has an application to Gromov’s mean dimension theory.

A Brody curve is a holomorphic map from the complex plane ℂ to a Hermitian manifold with bounded derivative. In this paper we study the value distribution of Brody curves from the viewpoint of moduli theory. The moduli space of Brody curves becomes infinite dimensional in general, and we study its “mean dimension”. We introduce the notion of “mean energy” and show that this can be used to estimate the mean dimension.

In this paper, using techniques of value distribution theory, we give a uniqueness theorem for meromorphic mappings of ℙm into ℙPn with (3n+1) moving targets and truncated multiplicities.

In this paper we shall establish some generalization of Nevanlinna’s Lemma on Logarithmic Derivative to the case of meromorphic maps from a finite analytic covering space over the m-dimensional complex affine space ℂm to a smooth complex projective variety. Then we shall apply this to “the Second Main Theorem” in Nevanlinna theory in several complex variables.

In this paper we first prove some criteria for the propagation of algebraic dependence of dominant meromorphic mappings from an analytic finite covering space X over the complex m-space into a projective algebraic manifold. We study this problem under a condition on the existence of meromorphic mappings separating the generic fibers of X. We next give applications of these criteria to the uniqueness problem of meromorphic mappings. We deduce unicity theorems for meromorphic mappings and also give some other applications. In particular, we study holomorphic mappings into a smooth elliptic curve E and give conditions under which two holomorphic mappings from X into E are algebraically related.