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

Entire holomorphic curves into projective spaces intersecting a generic hypersurface of high degree
[Courbes entières holomorphes dans un espace projectif intersectant une hypersurface générique de degré supérieur]

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, Tome 69 (2019) no. 2, pp. 653-671.

Résumé
Abstract

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.

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.

Reçu le : 2017-04-11
Révisé le : 2017-09-30
Accepté le : 2017-11-14
Publié le : 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

Affiliations des auteurs :

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

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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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, Tome 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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