On the hyperbolicity of general hypersurfaces

Damian Brotbek

doi:10.1007/s10240-017-0090-3

Damian Brotbek ¹

¹ Institut de Recherche Mathématique Avancée, Université de Strasbourg Strasbourg France

Publications Mathématiques de l'IHÉS, Volume 126 (2017), pp. 1-34

Abstract

In 1970, Kobayashi conjectured that general hypersurfaces of sufficiently large degree in $P^{n}$ are hyperbolic. In this paper we prove that a general sufficiently ample hypersurface in a smooth projective variety is hyperbolic. To prove this statement, we construct hypersurfaces satisfying a property which is Zariski open and which implies hyperbolicity. These hypersurfaces are chosen such that the geometry of their higher order jet spaces can be related to the geometry of a universal family of complete intersections. To do so, we introduce a Wronskian construction which associates a (twisted) jet differential to every finite family of global sections of a line bundle.

Received: 2016-06-21
Accepted: 2017-06-07
Online First: 2017-06-27
Published online: 2017-11-07

MR Zbl

DOI: 10.1007/s10240-017-0090-3

Author's affiliations:

Damian Brotbek ¹

¹ Institut de Recherche Mathématique Avancée, Université de Strasbourg Strasbourg France

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Damian Brotbek. On the hyperbolicity of general hypersurfaces. Publications Mathématiques de l'IHÉS, Volume 126 (2017), pp. 1-34. doi: 10.1007/s10240-017-0090-3

References
Cited by

[1.] O. Benoist Le théorème de Bertini en famille, Bull. Soc. Math. Fr., Volume 139 (2011), pp. 555-569 | DOI | Zbl | Numdam

[2.] G. Berczi, Towards the Green-Griffiths-Lang conjecture via equivariant localisation. ArXiv e-prints, 2015.

[3.] R. Brody Compact manifolds and hyperbolicity, Trans. Am. Math. Soc., Volume 235 (1978), pp. 213-219 | MR | Zbl

[4.] R. Brody; M. Green A family of smooth hyperbolic hypersurfaces in $P_{3}$ , Duke Math. J., Volume 44 (1977), pp. 873-874 | MR | DOI | Zbl

[5.] D. Brotbek Hyperbolicity related problems for complete intersection varieties, Compos. Math., Volume 150 (2014), pp. 369-395 | MR | DOI | Zbl

[6.] D. Brotbek Symmetric differential forms on complete intersection varieties and applications, Math. Ann., Volume 366 (2016), pp. 417-446 | MR | DOI | Zbl

[7.] D. Brotbek and L. Darondeau, Complete intersection varieties with ample cotangent bundles. ArXiv e-prints, 2015.

[8.] H. Clemens Curves on generic hypersurfaces, Ann. Sci. Éc. Norm. Super. (4), Volume 19 (1986), pp. 629-636 | DOI | MR | Zbl | Numdam

[9.] L. Darondeau On the logarithmic Green–Griffiths conjecture, Int. Math. Res. Not., Volume 2016 (2016), pp. 1871-1923 | MR | DOI | Zbl

[10.] L. Darondeau Slanted vector fields for jet spaces, Math. Z., Volume 282 (2016), pp. 547-575 | MR | DOI | Zbl

[11.] O. Debarre Varieties with ample cotangent bundle, Compos. Math., Volume 141 (2005), pp. 1445-1459 | MR | DOI | Zbl

[12.] J.-P. Demailly, Recent progress towards the Kobayashi and Green-Griffiths-Lang conjectures. Manuscript Institut Fourier. Available at: http://www-fourier.ujf-grenoble.fr/~demailly/preprints.html.

[13.] J.-P. Demailly Holomorphic Morse inequalities, Several Complex Variables and Complex Geometry, Part 2 (1991), pp. 93-114 | DOI

[14.] J.-P. Demailly Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Algebraic Geometry (1997), pp. 285-360

[15.] J.-P. Demailly Variétés hyperboliques et équations différentielles algébriques, Gaz. Math., Volume 73 (1997), pp. 3-23 | MR | Zbl

[16.] J.-P. Demailly Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, Pure Appl. Math. Q., Volume 7 (2011), pp. 1165-1207 (Special Issue: In memory of Eckart Viehweg) | MR | DOI | Zbl

[17.] J.-P. Demailly, Towards the Green-Griffiths-Lang conjecture. ArXiv e-prints, 2014.

[18.] J.-P. Demailly, Proof of the Kobayashi conjecture on the hyperbolicity of very general hypersurfaces. ArXiv e-prints, 2015.

[19.] J.-P. Demailly; J. El Goul Connexions méromorphes projectives partielles et variétés algébriques hyperboliques, C. R. Acad. Sci. Paris, Sér. I, Math., Volume 324 (1997), pp. 1385-1390 | MR | DOI | Zbl

[20.] J.-P. Demailly; J. El Goul Hyperbolicity of generic surfaces of high degree in projective 3-space, Am. J. Math., Volume 122 (2000), pp. 515-546 | MR | DOI | Zbl

[21.] Y. Deng, Effectivity in the Hyperbolicity-related problems. ArXiv e-prints, 2016.

[22.] Y. Deng, On the Diverio-Trapani Conjecture. ArXiv e-prints, 2017.

[23.] S. Diverio Differential equations on complex projective hypersurfaces of low dimension, Compos. Math., Volume 144 (2008), pp. 920-932 | MR | DOI | Zbl

[24.] S. Diverio Existence of global invariant jet differentials on projective hypersurfaces of high degree, Math. Ann., Volume 344 (2009), pp. 293-315 | MR | DOI | Zbl

[25.] S. Diverio; J. Merker; E. Rousseau Effective algebraic degeneracy, Invent. Math., Volume 180 (2010), pp. 161-223 | MR | DOI | Zbl

[26.] S. Diverio; E. Rousseau The exceptional set and the Green-Griffiths locus do not always coincide, Enseign. Math., Volume 61 (2015), pp. 417-452 | MR | DOI | Zbl

[27.] S. Diverio; S. Trapani A remark on the codimension of the Green-Griffiths locus of generic projective hypersurfaces of high degree, J. Reine Angew. Math., Volume 649 (2010), pp. 55-61 | MR | Zbl

[28.] L. Ein Subvarieties of generic complete intersections, Invent. Math., Volume 94 (1988), pp. 163-169 | MR | DOI | Zbl

[29.] L. Ein Subvarieties of generic complete intersections. II, Math. Ann., Volume 289 (1991), pp. 465-471 | MR | DOI | Zbl

[30.] J. El Goul Algebraic families of smooth hyperbolic surfaces of low degree in $P_{C}^{3}$ , Manuscr. Math., Volume 90 (1996), pp. 521-532 | MR | DOI | Zbl

[31.] G. Galbura Il wronskiano di un sistema di sezioni di un fibrato vettoriale di rango i sopra una curva algebrica ed il relativo divisore di Brill-Severi, Ann. Mat. Pura Appl. (4), Volume 98 (1974), pp. 349-355 | MR | DOI | Zbl

[32.] M. Green; P. Griffiths Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979) (1980), pp. 41-74 | DOI

[33.] J. Harris Algebraic Geometry (1995) (A first course, Corrected reprint of the 1992 original)

[34.] S. Kobayashi Hyperbolic Manifolds and Holomorphic Mappings (1970) | Zbl

[35.] S. Kobayashi Hyperbolic Complex Spaces (1998) | Zbl

[36.] J. Kollár Lectures on Resolution of Singularities (2007) | Zbl

[37.] D. Laksov Wronskians and Plücker formulas for linear systems on curves, Ann. Sci. Éc. Norm. Supér. (4), Volume 17 (1984), pp. 45-66 | DOI | Zbl | Numdam

[38.] S. Lang Hyperbolic and Diophantine analysis, Bull., New Ser., Am. Math. Soc., Volume 14 (1986), pp. 159-205 | MR | DOI | Zbl

[39.] K. Masuda; J. Noguchi A construction of hyperbolic hypersurface of $P^{n} (C)$ , Math. Ann., Volume 304 (1996), pp. 339-362 | MR | DOI | Zbl

[40.] M. McQuillan Diophantine approximations and foliations, Publ. Math. IHÉS, Volume 87 (1998), pp. 121-174 | MR | DOI | Zbl

[41.] M. McQuillan Holomorphic curves on hyperplane sections of 3-folds, Geom. Funct. Anal., Volume 9 (1999), pp. 370-392 | MR | DOI | Zbl

[42.] J. Merker Low pole order frames on vertical jets of the universal hypersurface, Ann. Inst. Fourier (Grenoble), Volume 59 (2009), pp. 1077-1104 | DOI | MR | Zbl | Numdam

[43.] J. Merker Algebraic differential equations for entire holomorphic curves in projective hypersurfaces of general type: Optimal lower degree bound, Geometry and Analysis on Manifolds (2015), pp. 41-142

[44.] A. M. Nadel Hyperbolic surfaces in $P^{3}$ , Duke Math. J., Volume 58 (1989), pp. 749-771 | MR | DOI | Zbl

[45.] M. Nakamaye Stable base loci of linear series, Math. Ann., Volume 318 (2000), pp. 837-847 | MR | DOI | Zbl

[46.] J. Noguchi Meromorphic mappings into a compact complex space, Hiroshima Math. J., Volume 7 (1977), pp. 411-425 | MR | Zbl

[47.] J. Noguchi Nevanlinna-Cartan theory over function fields and a Diophantine equation, J. Reine Angew. Math., Volume 487 (1997), pp. 61-83 | MR | Zbl

[48.] J. Noguchi Connections and the second main theorem for holomorphic curves, J. Math. Sci. Univ. Tokyo, Volume 18 (2011), pp. 155-180 | MR | Zbl

[49.] J. Noguchi; J. Winkelmann Nevanlinna Theory in Several Complex Variables and Diophantine Approximation (2014) | Zbl

[50.] G. Pacienza Subvarieties of general type on a general projective hypersurface, Trans. Am. Math. Soc., Volume 356 (2004), pp. 2649-2661 (electronic) | MR | DOI | Zbl

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

[52.] M. Păun Techniques de construction de différentielles holomorphes et hyperbolicité (d’après J.-P. Demailly, S. Diverio, J. Merker, E. Rousseau, Y.-T. Siu…), Astérisque, Volume 361 (2014), pp. 77-113 (Exp. No. 1061, vii) | Zbl | Numdam

[53.] E. Rousseau Équations différentielles sur les hypersurfaces de $P^{4}$ , J. Math. Pures Appl. (9), Volume 86 (2006), pp. 322-341 | MR | DOI | Zbl

[54.] E. Rousseau Weak analytic hyperbolicity of generic hypersurfaces of high degree in $P^{4}$ , Ann. Fac. Sci. Toulouse Math. (6), Volume 16 (2007), pp. 369-383 | DOI | MR | Zbl | Numdam

[55.] J. Greenlees Semple Some investigations in the geometry of curve and surface elements, Proc. Lond. Math. Soc., Volume 3 (1954), pp. 24-49 | MR | DOI | Zbl

[56.] B. Shiffman; M. Zaidenberg Hyperbolic hypersurfaces in $P^{n}$ of Fermat-Waring type, Proc. Am. Math. Soc., Volume 130 (2002), pp. 2031-2035 (electronic) | MR | DOI | Zbl

[57.] Y. Siu Noether-Lasker decomposition of coherent analytic subsheaves, Trans. Am. Math. Soc., Volume 135 (1969), pp. 375-385 | MR | DOI | Zbl

[58.] Y. Tong Siu Defect relations for holomorphic maps between spaces of different dimensions, Duke Math. J., Volume 55 (1987), pp. 213-251 | MR | DOI | Zbl

[59.] Y.-T. Siu Hyperbolicity in complex geometry, The Legacy of Niels Henrik Abel (2004), pp. 543-566 | DOI

[60.] Y.-T. Siu Hyperbolicity of generic high-degree hypersurfaces in complex projective space, Invent. Math., Volume 202 (2015), pp. 1069-1166 | MR | DOI | Zbl

[61.] Y.-T. Siu; S.-K. Yeung Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees, Am. J. Math., Volume 119 (1997), pp. 1139-1172 | MR | DOI | Zbl

[62.] C. Voisin On a conjecture of Clemens on rational curves on hypersurfaces, J. Differ. Geom., Volume 44 (1996), pp. 200-213 | MR | DOI | Zbl

[63.] S.-Y. Xie, On the ampleness of the cotangent bundles of complete intersections, ArXiv e-prints | arXiv

[64.] S.-Y. Xie, Generalized Brotbek’s symmetric differential forms and applications. ArXiv e-prints, 2016.

[65.] Z. Mikhail The complement to a general hypersurface of degree $2 n$ in ${CP}^{n}$ is not hyperbolic, Sib. Mat. Zh., Volume 28 (1987), pp. 91-100 (222) | MR

[66.] Z. Mikhail, Hyperbolic surfaces in $P^{3}$ : Examples, ArXiv Mathematics e-prints, 2003.

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