On the hyperbolicity of general hypersurfaces

Brotbek, Damian

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

In 1970, Kobayashi conjectured that general hypersurfaces of sufficiently large degree in Pn 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.

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

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

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

[4.] Brody, R.; Green, M. A family of smooth hyperbolic hypersurfaces in P3, Duke Math. J., Volume 44 (1977), pp. 873-874 | DOI | MR | Zbl

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

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

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

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

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

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

[11.] Debarre, O. Varieties with ample cotangent bundle, Compos. Math., Volume 141 (2005), pp. 1445-1459 | DOI | MR | 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.] Demailly, J.-P. Holomorphic Morse inequalities, Several Complex Variables and Complex Geometry, Part 2 (1991), pp. 93-114 | DOI

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

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

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

[20.] Demailly, J.-P.; El Goul, J. Hyperbolicity of generic surfaces of high degree in projective 3-space, Am. J. Math., Volume 122 (2000), pp. 515-546 | DOI | MR | 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.] Diverio, S. Differential equations on complex projective hypersurfaces of low dimension, Compos. Math., Volume 144 (2008), pp. 920-932 | DOI | MR | Zbl

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

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

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

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

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

[30.] El Goul, J. Algebraic families of smooth hyperbolic surfaces of low degree in PC3, Manuscr. Math., Volume 90 (1996), pp. 521-532 | DOI | MR | Zbl

[31.] Galbura, G. 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 | DOI | MR | Zbl

[32.] Green, M.; Griffiths, P. 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.] Harris, J. Algebraic Geometry (1995) (A first course, Corrected reprint of the 1992 original)

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

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

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

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

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

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

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

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

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

[43.] Merker, J. 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.] Nadel, A. M. Hyperbolic surfaces in P3, Duke Math. J., Volume 58 (1989), pp. 749-771 | DOI | MR | Zbl

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

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

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

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

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

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

[51.] Păun, 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

[52.] Păun, M. 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) | Numdam | Zbl

[53.] Rousseau, E. Équations différentielles sur les hypersurfaces de P4, J. Math. Pures Appl. (9), Volume 86 (2006), pp. 322-341 | DOI | MR | Zbl

[54.] Rousseau, E. Weak analytic hyperbolicity of generic hypersurfaces of high degree in P4, Ann. Fac. Sci. Toulouse Math. (6), Volume 16 (2007), pp. 369-383 | DOI | Numdam | MR | Zbl

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

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

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

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

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

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

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

[62.] Voisin, C. On a conjecture of Clemens on rational curves on hypersurfaces, J. Differ. Geom., Volume 44 (1996), pp. 200-213 | DOI | MR | 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.] Mikhail, Z. The complement to a general hypersurface of degree 2n in CPn is not hyperbolic, Sib. Mat. Zh., Volume 28 (1987), pp. 91-100 (222) | MR

[66.] Z. Mikhail, Hyperbolic surfaces in P3: Examples, ArXiv Mathematics e-prints, 2003.