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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $\\mathbb{P}^n$ be complex [[projective space|ZT9A6JK8QF]], and let $f \\colon \\C \\to \\mathbb{P}^n$ be a holomorphic curve.\nA *reduced representation* of $f$ is a map $F = (f_0, \\ldots, f_n) \\colon \\C \\to \\C^{n+1} \\setminus \\set{0}$ with [[entire|0G4F0XB9X2]] components and $f = [f_0 : \\cdots : f_n]$.\nIt exists, and it is unique up to multiplication by a nowhere-zero entire function.\nLet $\\norm{\\cdot}$ be the Euclidean norm on $\\C^{n+1}$ and let $r \u003e 0$.\nNevanlinna theory in several variables studies $f$ through the following functions of $r$.\n\n1. The *characteristic function* $T_f(r) = \\dfrac{1}{2\\pi} \\displaystyle\\int_0^{2\\pi} \\log \\norm{F(re^{i\\theta})} \\, d\\theta - \\log \\norm{F(0)}$.\n2. 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.\n3. The *proximity function* $m_f(r, H) = \\dfrac{1}{2\\pi} \\displaystyle\\int_0^{2\\pi} \\log \\frac{\\norm{F(re^{i\\theta})} \\, \\norm{a}}{\\abs{\\ip{F(re^{i\\theta})}{a}}} \\, d\\theta$.\n\nThe same notions exist for holomorphic maps from $\\C^m$ to $\\mathbb{P}^n$ or to projective varieties.\nThey use averages over spheres in $\\C^m$, and divisors in place of hyperplanes.\n\n## Remarks\n\nThe function $T_f$ does not depend on the reduced representation.\nIf $F$ is replaced by $gF$ with $g$ entire and nowhere zero, then $\\log\\abs{g}$ is harmonic, so its mean over the circle of radius $r$ is $\\log\\abs{g(0)}$.\nJensen's formula applied to $\\ip{F}{a}$ gives the *first main theorem* $T_f(r) = m_f(r, H) + N_f(r, H) + O(1)$ as $r \\to \\infty$.\n\nExample: let $g$ be a nonconstant entire function and $f = [1 : g] \\colon \\C \\to \\mathbb{P}^1$, with $F = (1, g)$.\nThen $T_f(r) = \\frac{1}{2\\pi}\\int_0^{2\\pi} \\log\\sqrt{1 + \\abs{g(re^{i\\theta})}^2}\\, d\\theta + O(1)$.\nSince $\\log^+ \\abs{w} \\le \\log\\sqrt{1 + \\abs{w}^2} \\le \\log^+\\abs{w} + \\frac12 \\log 2$, this differs from the one-variable [[Nevanlinna characteristic|YT6Z0GWS0W]] $T(r, g)$ by a bounded amount.\n\nNon-example: if $f(\\C) \\subset H$, then $\\ip{F}{a} \\equiv 0$ and the counting and proximity functions of $H$ are not defined.\n\nHyperplanes $H_1, \\ldots, H_q$ are in *general position* if any $n + 1$ of their coefficient vectors are [[linearly independent|0CRMGPAH83]].\nCartan's second main theorem concerns a linearly nondegenerate $f$, that is, $f(\\C)$ lies in no hyperplane, and hyperplanes $H_1, \\ldots, H_q$ in general position.\nIt states that $(q - n - 1)\\, T_f(r) \\le \\sum_{j=1}^q N_f(r, H_j) + O(\\log T_f(r) + \\log r)$.\nThe inequality holds for all $r$ outside a set of finite Lebesgue measure.\n",
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