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-Viet³ ; Xie, Song-Yan⁴

¹ Department of Mathematics Graduate School of Science, Osaka University Toyonaka, Osaka 560-0043 (Japan)
² Department of Mathematics College of Education, Hue University 34 Le Loi St. Hue City (Vietnam)
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.

Abstract (VO)

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

$$T_f\left(r\right)\le N_f^{\left[1\right]}(r,D)+O\left(log{T}_f\left(r\right)\right)+\delta logr,$$

where $T_f\left(r\right)$ and $N_f^{\left[1\right]}(r,D)$ 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.

Résumé

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

$$T_f\left(r\right)\le N_f^{\left[1\right]}(r,D)+O\left(log{T}_f\left(r\right)\right)+\delta logr,$$

où $T_f\left(r\right)$ et $N_f^{\left[1\right]}(r,D)$ 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.

Détail

Reçu le : 2017-04-11
Révisé le : 2017-09-30
Accepté le : 2017-11-14
Publié le : 2019-06-03

DOI : 10.5802/aif.3253

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

Licence : CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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