# ANNALES DE L'INSTITUT FOURIER

Entire holomorphic curves into projective spaces intersecting a generic hypersurface of high degree
Annales de l'Institut Fourier, to appear, 19 p.

In this note, we establish the following Second Main Theorem type estimate for every algebraically nondegenerate entire curve $f:ℂ\to {ℙ}^{n}\left(ℂ\right)$, in presence of a generic divisor $D\subset {ℙ}^{n}\left(ℂ\right)$ of sufficiently high degree $d\ge 15\left(5n+1\right){n}^{n}$: for every $r$ outside a subset of $ℝ$ of finite Lebesgue measure and every real positive constant $\delta$, we have

${T}_{f}\left(r\right)\le {N}_{f}^{\left[1\right]}\left(r,D\right)+O\left(log{T}_{f}\left(r\right)\right)+\delta logr,$

where ${T}_{f}\left(r\right)$ and ${N}_{f}^{\left[1\right]}\left(r,D\right)$ stand for the order function and the $1$-truncated counting function in Nevanlinna theory. This inequality quantifies recent results on the logarithmic Green–Griffiths conjecture.

Dans cet article, nous établissons le théorème suivant : pour toute courbe entière algébriquement non-dégénérée $f:ℂ\to {ℙ}^{n}\left(ℂ\right)$ intersectant un diviseur générique $D\subset {ℙ}^{n}\left(ℂ\right)$ de degré $d\ge 15\left(5n+1\right){n}^{n}$ pour tous $r$ en dehors d’un sous-ensemble de $ℝ$ de mesure de Lebesgue finie et toute constante réelle positive $\delta$, on a

${T}_{f}\left(r\right)\le {N}_{f}^{\left[1\right]}\left(r,D\right)+O\left(log{T}_{f}\left(r\right)\right)+\delta logr,$

${T}_{f}\left(r\right)$ et ${N}_{f}^{\left[1\right]}\left(r,D\right)$ sont la fonction d’ordre et la fonction de comptage $1$-tronqué dans la théorie de Nevanlina. Cette inégalité quantifie des résultats récents sur la conjecture de Green–Griffiths logarithmique.

Revised : 2017-09-30
Accepted : 2017-11-14
Classification:  32H30,  32A22,  30D35,  32Q45
Keywords: Nevanlinna theory, Second Main Theorem, holomorphic curve, Green–Griffiths’ conjecture, algebraic degeneracy
@unpublished{AIF_0__0_0_A17_0,
author = {Huynh, Dinh Tuan and Vu, Duc-Viet and Xie, Song-Yan},
title = {Entire holomorphic curves into projective spaces intersecting a generic hypersurface of high degree},
note = {to appear in \emph{Annales de l'Institut Fourier}},
}

Huynh, Dinh Tuan; Vu, Duc-Viet; Xie, Song-Yan. Entire holomorphic curves into projective spaces intersecting a generic hypersurface of high degree. Annales de l'Institut Fourier, to appear, 19 p.

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