Low pole order frames on vertical jets of the universal hypersurface
[Repères méromorphes sur les jets de l’hypersurface universelle]
Annales de l'Institut Fourier, Tome 59 (2009) no. 3, pp. 1077-1104.

Pour des ordres de jets petits, on sait construire des repères méromorphes sur l’espace des jets verticaux J vert k (𝒳) de l’hypersurface universelle

𝒳n+1×(n+1+d)!(n+1)!d!-1

qui paramétrise toutes les hypersurfaces projectives X n+1 () de degré d. Siu a annoncé en 2004 que, pour k=n, il existe deux constantes c n 1 et c n 1 telles que le fibré tangent tensorisé

TJvertn(𝒳)𝒪n+1(cn)𝒪(n+1+d)!((n+1)!d!)-1(cn)

est engendré par ses sections globales. Nous établissons cette propriété hors d’un certain ensemble algébrique exceptionnel ΣJ vert n (𝒳) défini par l’annulation de certains wronskiens, avec l’ordre de pôles effectif c n =1 2(n 2 +5n), retrouvant ainsi c 2 =7 (Paŭn), c 3 =12 (Rousseau), et avec c n =1.

De plus, quitte à augmenter c n jusqu’à c n =n 2 +2n, la même propriété d’engendrement est satisfaite hors du plus petit sous-ensemble Σ ˜ΣJ vert n (𝒳) qui est défini par l’annulation de tous les jets d’ordre 1. Des applications à la dégénérescence algébrique faible (avec Σ) et forte (avec Σ ˜) des courbes holomorphes entières X en découleront prochainement.

For low order jets, it is known how to construct meromorphic frames on the space of the so-called vertical k-jets J vert k (𝒳) of the universal hypersurface

𝒳n+1×(n+1+d)!((n+1)!d!)-1

parametrizing all projective hypersurfaces X n+1 () of degree d. In 2004, for k=n, Siu announced that there exist two constants c n 1 and c n 1 such that the twisted tangent bundle

TJvertn(𝒳)𝒪n+1(cn)𝒪(n+1+d)!((n+1)!d!)-1(cn)

is generated at every point by its global sections. In the present article, we establish this property outside a certain exceptional algebraic subset ΣJ vert n (𝒳) defined by the vanishing of certain Wronskians, with the effective pole order c n =1 2(n 2 +5n), thus recovering c 2 =7 (Paŭn), c 3 =12 (Rousseau), and with c n =1.

Moreover, at the cost of raising c n up to c n =n 2 +2n, the same generation property holds outside the smaller set Σ ˜ΣJ vert n (𝒳) which is defined by the vanishing of all first order jets. Applications to weak (with Σ) and to strong (with Σ ˜) algebraic degeneracy of entire holomorphic curves X are upcoming.

DOI : 10.5802/aif.2458
Classification : 32Q45, 14N05, 14J70
Keywords: Multivariate Faà di Bruno formula, projective algebraic hypersurfaces, jets of holomorphic curves, weak and strong Green-Griffiths algebraic degeneracy
Mot clés : formule de Faà di Bruno à plusieurs variables, hypersurfaces projectives algébriques, jets de courbes holomorphes, dégénérescence algébrique faible et forte au sens de Green-Griffiths
Merker, Joël 1

1 tabacckludge ’Ecole Normale Supérieure UMR 8553 du CNRS Département de Mathématiques et Applications 45 rue d’Ulm 75230 Paris Cedex 05 (France)
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Merker, Joël. Low pole order frames on vertical jets of the universal hypersurface. Annales de l'Institut Fourier, Tome 59 (2009) no. 3, pp. 1077-1104. doi : 10.5802/aif.2458. https://aif.centre-mersenne.org/articles/10.5802/aif.2458/

[1] Clemens, H. Curves on generic hypersurfaces, Ann. Sci. École Norm. Sup., Volume 19 (1986), pp. 629-636 | Numdam | MR | Zbl

[2] Constantine, G. M.; Savits, T. H. A multivariate Faà di Bruno formula with applications, Trans. Amer. Math. Soc., Volume 348 (1996), pp. 503-520 | DOI | MR | Zbl

[3] Demailly, J.-P. Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Proc. Sympos. Pure Math. Amer. Math. Soc., Volume 62 (1997), pp. 285-360 | MR | Zbl

[4] Diverio, S. Existence of global invariant jet differentials on projective hypersurfaces of high degree (to appear in Math. Ann., arxiv.org/abs/0802.0045/)

[5] Diverio, S. Differential equations on complex projective hypersurfaces of low dimension, Compos. Math., Volume 144 (2008), pp. 920-932 (arxiv.org/abs/0706.1051/) | DOI | MR

[6] Diverio, S.; Merker, J.; Rousseau, E. Effective algebraic degeneracy (nov. 2008) (preprint, arxiv.org/abs/0811.2346/)

[7] Duval, J. Sur le lemme de Brody, Invent. Math., Volume 173 (2008), pp. 305-314 | DOI | MR | Zbl

[8] Ein, L. Subvarieties of generic complete intersections, Invent. Math., Volume 94 (1988), pp. 163-169 II. Math. Ann. 289 (1991), p.465–471 | DOI | MR | Zbl

[9] Green, M.; Griffiths, P. Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979, Proc. Inter. Sympos. Berkeley, CA, 1979 (1980), pp. 41-74 | MR | Zbl

[10] Merker, J. An algorihm to generate all polynomials in the k -jet of a holomorphic disc D n that are invariant under source reparametrization (arxiv.org/abs/0808.3547/)

[11] Merker, J. Four explicit formulas for the prolongations of an infinitesimal Lie symmetry and multivariate Faà di Bruno formulas (arxiv.org/abs/math/0411650)

[12] Merker, J. Jets de Demailly-Semple d’ordres 4 et 5 en dimension 2, Int. J. Contemp. Math. Sciences, Volume 3 (2008), pp. 861-933 (arxiv.org/abs/0710.2393) | MR

[13] Paŭn, M. Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity, Math. Ann., Volume 340 (2008), pp. 875-892 | DOI | MR | Zbl

[14] Rousseau, E. Weak analytic hyperbolicity of complements of generic surfaces of high degree in projective 3-space, Osaka J. Math., Volume 44 (2007), pp. 955-971 | MR | Zbl

[15] Rousseau, E. Weak analytic hyperbolicity of generic hypersurfaces of high degree in 4 , Ann. Fac. Sci. Toulouse, Volume XIV (2007), pp. 369-383 | DOI | Numdam | MR | Zbl

[16] Rousseau, Erwan Équations différentielles sur les hypersurfaces de  4 , J. Math. Pures Appl. (9), Volume 86 (2006), pp. 322-341 | MR | Zbl

[17] Rousseau, Erwan Étude des jets de Demailly-Semple en dimension 3, Ann. Inst. Fourier, Volume 56 (2006), pp. 397-421 | DOI | Numdam | MR | Zbl

[18] Siu, Y.-T. Hyperbolicity in complex geometry, p.543-566, The legacy of Niels Henrik Abel, Springer-Verlag, Berlin, 2004 | MR | Zbl

[19] Trapani, S. Numerical criteria for the positivity of the difference of ample divisors, Math. Z., Volume 219 (1995), pp. 387-401 | DOI | MR | Zbl

[20] Voisin, C. On a conjecture of Clemens on rational curves on hypersurfaces, J. Diff. Geom., Volume 44 (1996), pp. 200-213 A correction: “On a conjecture of Clemens on rational curves on hypersurfaces”, J. Diff. Geom. 49 (1998), p.601–611 | MR | Zbl

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