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, à paraître, 19 p.

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

Tf(r)Nf[1](r,D)+OlogTf(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

Tf(r)Nf[1](r,D)+OlogTf(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.

Reçu le : 2017-04-11
Révisé le : 2017-09-30
Accepté le : 2017-11-14
Publié le : 2019-03-08
Classification:  32H30,  32A22,  30D35,  32Q45
Mots clés: théorie de Nevanlina, le deuxième théorème fondamental, conjecture de Green–Griffiths, dégénéré algébriquement
@unpublished{AIF_0__0_0_A17_0,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, à paraître, 19 p.

[1] An, Do Phuong; Quang, Si Duc; Thai, Do Duc The second main theorem for meromorphic mappings into a complex projective space, Acta Math. Vietnam., Tome 38 (2013) no. 1, pp. 187-205 | Article | MR 3098207 | Zbl 1375.32033

[2] Berczi, Gergely Towards the Green-Griffiths-Lang conjecture via equivariant localisation (2015) (https://arxiv.org/abs/1509.03406 )

[3] Cartan, Henri Sur les zéros des combinaisons linéaires de p fonctions holomorphesdonnées, Mathematica, Tome 7 (1933), pp. 80-103

[4] Constantine, Gregory M.; Savits, Thomas H. A multivariate Faà di Bruno formula with applications, Trans. Am. Math. Soc., Tome 348 (1996) no. 2, pp. 503-520 | Article | MR 1325915 | Zbl 0846.05003

[5] Darondeau, Lionel On the logarithmic Green-Griffiths conjecture, Int. Math. Res. Not. (2016) no. 6, pp. 1871-1923 | Article | MR 3509943

[6] Darondeau, Lionel Slanted vector fields for jet spaces, Math. Z., Tome 282 (2016) no. 1-2, pp. 547-575

[7] Demailly, Jean-Pierre Variétés projectives hyperboliques et équations différentielles algébriques, Journée en l’Honneur de Henri Cartan, Société Mathématique de France (SMF Journée Annuelle) Tome 1997 (1997), pp. 3-17 | MR 1492596 | Zbl 0937.32012

[8] Demailly, Jean-Pierre Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, Pure Appl. Math. Q., Tome 7 (2011) no. 4, pp. 1165-1207 | Article | MR 2918158 | Zbl 1316.32014

[9] Demailly, Jean-Pierre Hyperbolic algebraic varieties and holomorphic differential equations, Acta Math. Vietnam., Tome 37 (2012) no. 4, pp. 441-512 | MR 3058660

[10] Dethloff, Gerd-Eberhard; Lu, Steven Shin-Yi Logarithmic jet bundles and applications, Osaka J. Math., Tome 38 (2001) no. 1, pp. 185-237 | MR 1824906

[11] Diverio, Simone; Merker, Joël; Rousseau, Erwan Effective algebraic degeneracy, Invent. Math., Tome 180 (2010), pp. 161-223

[12] Erëmenko, Alexandre È.; Sodin, Mikhail L. Distribution of values of meromorphic functions and meromorphic curves from the standpoint of potential theory, Algebra Anal., Tome 3 (1991) no. 1, pp. 131-164 | MR 1120844

[13] Green, Mark; Griffiths, Phillip Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, 1979), Springer (1980), pp. 41-74 | MR 609557 | Zbl 0508.32010

[14] Iitaka, Shigeru Algebraic geometry. An introduction to birational geometry of algebraic varieties, Springer, Graduate Texts in Mathematics, Tome 76 (1982), x+357 pages | MR 637060 | Zbl 0491.14006

[15] Merker, Joël Low pole order frames on vertical jets of the universal hypersurface, Ann. Inst. Fourier, Tome 59 (2009) no. 3, pp. 1077-1104 | MR 2543663

[16] Merker, Joël Algebraic differential equations for entire holomorphic curves in projective hypersurfaces of general type: optimal lower degree bound, Geometry and analysis on manifolds, Birkhäuser/Springer (Progress in Mathematics) Tome 308 (2015), pp. 41-142 | Article | MR 3331396 | Zbl 1333.32033

[17] Mumford, David Abelian varieties, Oxford University Press, Tata Institute of Fundamental Research Studies in Mathematics, Tome 5 (1970), viii+242 pages

[18] Noguchi, Junjiro Logarithmic jet spaces and extensions of de Franchis’ theorem, Contributions to several complex variables, Friedr. Vieweg & Sohn (Aspects of Mathematics) Tome E9 (1986), pp. 227-249 | MR 859200 | Zbl 0598.32021

[19] Noguchi, Junjiro; Winkelmann, Jörg Nevanlinna theory in several complex variables and Diophantine approximation, Springer, Grundlehren der Mathematischen Wissenschaften, Tome 350 (2014), xiv+416 pages | Article | MR 3156076

[20] Noguchi, Junjiro; Winkelmann, Jörg; Yamanoi, Katsutoshi The second main theorem for holomorphic curves into semi-abelian varieties, Acta Math., Tome 188 (2002) no. 1, pp. 129-161 | Article | MR 1947460

[21] Noguchi, Junjiro; Winkelmann, Jörg; Yamanoi, Katsutoshi The second main theorem for holomorphic curves into semi-abelian varieties. II, Forum Math., Tome 20 (2008) no. 3, pp. 469-503 | MR 2418202

[22] Păun, Mihai; Sibony, Nessim Value Distribution Theory for Parabolic Riemann Surfaces (2014) (https://arxiv.org/abs/1403.6596 )

[23] Ru, Min A defect relation for holomorphic curves intersecting hypersurfaces, Am. J. Math., Tome 126 (2004) no. 1, pp. 215-226 | MR 2033568

[24] Ru, Min Holomorphic curves into algebraic varieties, Ann. Math., Tome 169 (2009), pp. 255-267

[25] Siu, Yum-Tong Hyperbolicity in complex geometry, The legacy of Niels Henrik Abel, Springer (2004), pp. 543-566 | MR 2077584

[26] Siu, Yum-Tong Hyperbolicity of generic high-degree hypersurfaces in complex projective space, Invent. Math., Tome 202 (2015) no. 3, pp. 1069-1166 | Article | MR 3425387

[27] Thai, Do Duc; Vu, Duc-Viet Holomorphic mappings into compact complex manifolds, Houston J. Math., Tome 43 (2017) no. 3, pp. 725-762 | Zbl 1393.32007

[28] Yamanoi, Katsutoshi Holomorphic curves in abelian varieties and intersections with higher codimensional subvarieties, Forum Math., Tome 16 (2004) no. 5, pp. 749-788 | MR 2096686