Entire holomorphic curves into projective spaces intersecting a generic hypersurface of high degree

Huynh, Dinh Tuan; Vu, Duc-Viet; Xie, Song-Yan

doi:10.5802/aif.3253

Huynh, Dinh Tuan ^1,²; Vu, Duc-Viet ³; Xie, Song-Yan ⁴

¹ Department of Mathematics College of Education, Hue University 34 Le Loi St. Hue City (Vietnam)
² Department of Mathematics Graduate School of Science, Osaka University Toyonaka, Osaka 560-0043 (Japan)
³ Korea institute for advanced study 85 Hoegiro, Dongdaemun-gu Seoul 02455 (Republic of Korea)
⁴ Max-Planck-Institut für Mathematik, Vivatsgasse 7 53111 Bonn (Germany)

Annales de l'Institut Fourier, Volume 69 (2019) no. 2, pp. 653-671.

Abstract
Résumé

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

T_{f} (r) \leq N_{f}^{[1]} (r, D) + O (log T_{f} (r)) + δ log r,

where $T_{f} (r)$ and $N_{f}^{[1]} (r, D)$ 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 $f : ℂ \to ℙ^{n} (ℂ)$ intersectant un diviseur générique $D \subset ℙ^{n} (ℂ)$ de degré $d \geq 15 (5 n + 1) n^{n}$ pour tous $r$ en dehors d’un sous-ensemble de $ℝ$ de mesure de Lebesgue finie et toute constante réelle positive $δ$ , on a

T_{f} (r) \leq N_{f}^{[1]} (r, D) + O (log T_{f} (r)) + δ log r,

où $T_{f} (r)$ et $N_{f}^{[1]} (r, D)$ 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.

Received: 2017-04-11
Revised: 2017-09-30
Accepted: 2017-11-14
Published online: 2019-06-03

Zbl

DOI: 10.5802/aif.3253

Classification: 32H30, 32A22, 30D35, 32Q45
Keywords: Nevanlinna theory, Second Main Theorem, holomorphic curve, Green–Griffiths’ conjecture, algebraic degeneracy
Mot clés : théorie de Nevanlina, le deuxième théorème fondamental, conjecture de Green–Griffiths, dégénéré algébriquement

Author's affiliations:

Huynh, Dinh Tuan ^{1, 2}; Vu, Duc-Viet ³; Xie, Song-Yan ⁴

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Entire holomorphic curves into projective spaces intersecting a generic hypersurface of high degree},
     journal = {Annales de l'Institut Fourier},
     pages = {653--671},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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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, Volume 69 (2019) no. 2, pp. 653-671. doi : 10.5802/aif.3253. https://aif.centre-mersenne.org/articles/10.5802/aif.3253/

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