Nevanlinna theory in several variables

Definition

Let $n \ge 1$, let $\mathbb{P}^n$ be complex projective space, and let $f \colon \C \to \mathbb{P}^n$ be a holomorphic curve. A reduced representation of $f$ is a map $F = (f_0, \ldots, f_n) \colon \C \to \C^{n+1} \setminus \set{0}$ with entire components and $f = [f_0 : \cdots : f_n]$. It exists, and it is unique up to multiplication by a nowhere-zero entire function. Let $\norm{\cdot}$ be the Euclidean norm on $\C^{n+1}$ and let $r > 0$. Nevanlinna theory in several variables studies $f$ through the following functions of $r$.

1. The characteristic function $T_f(r) = \dfrac{1}{2\pi} \displaystyle\int_0^{2\pi} \log \norm{F(re^{i\theta})} \, d\theta - \log \norm{F(0)}$.
2. For a hyperplane $H = \set{a_0 x_0 + \cdots + a_n x_n = 0}$ with $a = (a_0, \ldots, a_n) \ne 0$ and $f(\C) \not\subset H$, write $\ip{F}{a} = a_0 f_0 + \cdots + a_n f_n$. The counting function $N_f(r, H) = \displaystyle\int_0^r \frac{n(t) - n(0)}{t} \, dt + n(0) \log r$, where $n(t)$ is the number of zeros of $\ip{F}{a}$ in $\set{\abs{z} \le t}$, counted with multiplicity.
3. The proximity function $m_f(r, H) = \dfrac{1}{2\pi} \displaystyle\int_0^{2\pi} \log \frac{\norm{F(re^{i\theta})} \, \norm{a}}{\abs{\ip{F(re^{i\theta})}{a}}} \, d\theta$.

The same notions exist for holomorphic maps from $\C^m$ to $\mathbb{P}^n$ or to projective varieties. They use averages over spheres in $\C^m$, and divisors in place of hyperplanes.