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

Dans cet article, nous établissons le théorème suivant : pour toute courbe entière algébriquement non-dégénérée f: n () intersectant un diviseur générique D n () de degré d15(5n+1)n n pour tous r en dehors d’un sous-ensemble de de mesure de Lebesgue finie et toute constante réelle positive δ, on a

T f (r)N f [1] (r,D)+Olog T f (r)+δlogr,

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: n (), in presence of a generic divisor D n () of sufficiently high degree d15(5n+1)n n : for every r outside a subset of of finite Lebesgue measure and every real positive constant δ, we have

T f (r)N f [1] (r,D)+Olog T f (r)+δlogr,

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.

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

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)
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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. http://archive.numdam.org/articles/10.5802/aif.3253/

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