Abstract
In this paper, we prove that in any projective manifold, the complements of general hypersurfaces of sufficiently large degree are Kobayashi hyperbolic. We also provide an effective lower bound on the degree. This confirms a conjecture by S. Kobayashi in 1970. Our proof, based on the theory of jet differentials, is obtained by reducing the problem to the construction of a particular example with strong hyperbolicity properties. This approach relies the construction of higher order logarithmic connections allowing us to construct logarithmic Wronskians. These logarithmic Wronskians are the building blocks of the more general logarithmic jet differentials we are able to construct. As a byproduct of our proof, we prove a more general result on the orbifold hyperbolicity for generic geometric orbifolds in the sense of Campana, with only one component and large multiplicities. We also establish a Second Main Theorem type result for holomorphic entire curves intersecting general hypersurfaces, and we prove the Kobayashi hyperbolicity of the cyclic cover of a general hypersurface, again with an explicit lower bound on the degree of all these hypersurfaces.
Similar content being viewed by others
Abbreviations
- \(f:({{\mathbb {C}}},0)\rightarrow X\) :
-
Germ of holomorphic curve
- \(J_kX\rightarrow X\) :
-
Fiber bundle of k-jets of germs of holomorphic curves in X
- \(j_kf\in J_kX\) :
-
k-jet of germ of curve f in \(J_kX\)
- \(J_k(X,\log D)\) :
-
Fiber bundle of logarithmic k-jets of germs of holomorphic curves in X
- \(E_{k,m}^\mathrm{GG}\Omega _X, E_{k,m}^{\mathrm{GG}}\Omega _X(\log D)\) :
-
Green–Griffiths bundle of (logarithmic) jet differentials of order k and weight m
- (X, V):
-
Directed manifolds
- \(X_k\) :
-
Demailly–Semple k-jet tower of the directed manifold (X, V)
- (X, D, V):
-
Logarithmic directed manifold
- \(E_{k,m}\Omega _X, E_{k,m} \Omega _X(\log D)\) :
-
Invariant (logarithmic) jet bundle of order k and weight m
- \(X_k(D)\) :
-
logarithmic Demailly(–Semple) k-jet tower associated to logarithmic directed manifold \(\big (X,D,T_X(-\log D)\big )\)
- \(\pi _{0,k}:X_k(D)\rightarrow X\) :
-
The natural projection map
- \(J^kL\) :
-
Jet bundle of a line bundle L (1.14)
- \(j_L^ks \in H^0(X,J^kL)\) :
-
k-jet of the holomorphic section \(s\in H^0(X,L)\)
- \(W_L(\bullet )\) :
-
Wronskian (2.1) associated to the line bundle L
- \(W_{D}(\bullet )\) :
-
Logarithmic Wronskian associated to log pair (X, D) (2.6)
- \(\omega _{D}(\bullet )\) :
-
\(({\pi }_{0,k})_*\omega _{D}(\bullet )=W_{D}(\bullet )\) (2.8)
- \( {\nabla }_{ \mathfrak {U}}^j(\bullet )\) :
-
Higher order logarithmic connection in the trivialization tower \(\mathfrak {U}\) (2.11)
- \(\omega _{D}^{\prime }(\bullet )\) :
- \( {\mathfrak {w}}_{{X}_k(D)}\) :
-
k-th logarithmic Wronskian ideal sheaf of the log manifold (X, D)
- \(\mathbb {L}\) :
-
The total space of the line bundle \(A^{\otimes m}\) on Y
- \(W_{{\mathbb {L}},Y} (\bullet )\) :
-
Logarithmic Wronskian associated to the log pair \((\mathbb {L},Y)\)
- \(\mathbb {L}_k\) :
-
Log Demailly k-jet tower of log directed manifold \(\big (\mathbb {L},Y,T_{\mathbb {L}}(-\log Y)\big )\)
- \(\omega _{\log }(\bullet )\) :
-
\(({\pi }_{0,k})_*\omega _{\log } (\bullet )=W_{{\mathbb {L}},Y} (\bullet )\)
- \({\mathfrak {w}}_{k,{\mathbb {L}},Y}\), \({\mathfrak {w}}^{\prime }_{k,{\mathbb {L}},Y}\) :
-
Ideal sheaves of \(\mathbb {L}_k\) in (2.20)
- \(\nu _k:\widetilde{{\mathbb {L}}}_k\rightarrow \mathbb {L}_k\) :
-
The blow-up of \({\mathfrak {w}}_{k,{\mathbb {L}},Y}\) and \({\mathfrak {w}}^{\prime }_{k,{\mathbb {L}},Y}\)
- \((H_\sigma ,D_\sigma )\) :
-
The sub-log manifold of the log pair \((\mathbb {L},Y)\) induced by \(\sigma \in H^0(Y,A^m)\)
- \( H_{\sigma ,k}\) :
-
Log Demailly tower of log directed manifold \(\big (H_\sigma ,D_\sigma ,T_{H_\sigma }(-\log D_\sigma )\big )\)
- \(\mu _{\sigma ,k}:\widetilde{H}_{\sigma ,k}\rightarrow H_{\sigma ,k}\) :
-
The blow-up of \({\mathfrak {w}}_{H_{\sigma ,k}}\)
- \(\mathbf {a}\in \mathbb {A}\) :
-
Family of hypersurfaces in Y parametrized by certain Fermat-type hypersurfaces defined in Section 3.1
- \( (\mathscr {H},\mathscr {D})\rightarrow \mathbb {A}_\mathrm{sm}\) :
-
Smooth family of sub-log pairs of \((\mathbb {L},Y)\) induced by Fermat-type hypersurfaces
- \(\mathscr {H}_k\) :
-
Log Demailly k-jet tower of log directed manifold \(\big (\mathscr {H},\mathscr {D},T_{\mathscr {H}/\mathbb {A}_\mathrm{sm}}(-\log \mathscr {D})\big )\)
- \(\omega _{\log ,I_1,\ldots ,I_k}(\bullet )\) :
-
Modified logarithmic Wronskians (3.2)
- \(E^\mathrm{GG}_{k,N}\Omega _{Y,\Delta }\) :
-
Orbifold jet differentials of order k and weight N associated to Campana orbifold \((Y,\Delta )\).
References
O. Benoist. Le théorème deBertini en famille. Bulletin de la Société Mathématique de France, (4)139 (2011), 555–569
R. Brody. Compact manifolds and hyperbolicity. Transactions of the American Mathematical Society 235 (1978), 213–219. https://doi.org/10.2307/1998216
D. Brotbek. On the hyperbolicity of general hypersurfaces. Publications Mathématiques de l’IHÉS, 126 (2017), 1–34. https://doi.org/10.1007/s10240-017-0090-3
D. Brotbek and Y. Deng. On the positivity of the logarithmic cotangent bundle. To appear in Annales de l’Institut Fourier (en l’honneur du professeur Jean-Pierre Demailly) (2017). arXiv:1712.09887
D. Brotbek and L. Darondeau. Complete intersection varieties with ample cotangent bundles. Inventiones Mathematicae, (3)212 (2018), 913–940. https://doi.org/10.1007/s00222-017-0782-9.
F. Campana. Orbifolds, special varieties and classification theory. Annales de l’Institut Fourier (Grenoble), (3)54 (2004), 499–630. http://aif.cedram.org/item?id=AIF_2004__54_3_499_0
F. Campana, L. Darondeau, and E. Rousseau. Orbifold hyperbolicity. arXiv e-prints (2018). arXiv:1803.10716
L. Darondeau. Fiber integration on the Demailly tower. Annales de l’Institut Fourier (Grenoble), (1)66 (2016), 29–54. http://aif.cedram.org/item?id=AIF_2016__66_1_29_0
L. Darondeau. On the logarithmic Green–Griffiths conjecture. International Mathematics Research Notices IMRN, (6) (2016), 1871–1923. https://doi.org/10.1093/imrn/rnv078
L. Darondeau. Slanted vector fields for jet spaces. Mathematische Zeitschrift, (1-2)282 (2016), 547–575. https://doi.org/10.1007/s00209-015-1553-1
J.-P. Demailly. Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. In: Algebraic Geometry—SantaCruz 1995, Proceedings of Symposia in Pure Mathematics, Vol. 62, American Mathematical Society, Providence, RI (1997), pp. 285–360.
J.-P. Demailly. Recent results on the Kobayashi and the Green–Griffiths–Lang conjectures. arXiv e-prints (2018). arXiv:1801.04765
Y. Deng. On the Diverio–Trapani Conjecture, to appear in Annales Scientifiques de l’Ecole Normale Supérieure (2017). arXiv:1703.07560
G.-E. Dethloff and S.S.-Y. Lu. Logarithmic jet bundles and applications. The Osaka Journal of Mathematics, (1)38 (2001), 185–237. http://projecteuclid.org/euclid.ojm/1153492319.
S. Diverio. Existence of global invariant jet differentials on projective hypersurfaces of high degree. Mathematische Annalen, (2)344 (2009), 293–315. https://doi.org/10.1007/s00208-008-0306-4.
S. Diverio, J. Merker, and E. Rousseau. Effective algebraic degeneracy. Inventiones Mathematicae (1)180 (2010), 161–223. https://doi.org/10.1007/s00222-010-0232-4
J. El Goul. Logarithmic jets and hyperbolicity. The Osaka Journal of Mathematics, (2)40 (2003), 469–491. http://projecteuclid.org/euclid.ojm/1153493095
H. Fujimoto. On holomorphic maps into a taut complex space. The Nagoya Mathematical Journal, 46 (1972), 49–61. http://projecteuclid.org/euclid.nmj/1118798592
M.L. Green. The hyperbolicity of the complement of \(2n+1\) hyperplanes in general position in \(\mathbb{P}_{n}\) and related results. Proceedings of the American Mathematical Society, (1)66 (1977), 109–113. https://doi.org/10.2307/2041540
A. Grothendieck. Éléments de Géométrie Algébrique. IV. étude locale des schémas et des morphismes de schémas. III. Publications Mathématiques de l’IHÉS (28)(1966), 255. http://www.numdam.org/item?id=PMIHES_1966__28__255_0
R. Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics, Vol. 52. Springer-Verlag, New York–Heidelberg (1977).
D.T. Huynh, D.-V. Vu, and S.-Y. Xie. Entire holomorphic curves into projective spaces intersecting a generic hypersurface of high degree. To appear in Annales de l’Institut Fourier (2017). arXiv:1704.03358 [math.AG]
S. Kobayashi. Hyperbolic Manifolds and Holomorphic Mappings. Pure and Applied Mathematics, Vol. 2, Marcel Dekker Inc., New York (1970).
S. Kobayashi. Hyperbolic Complex Spaces. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 318, Springer-Verlag, Berlin (1998). https://doi.org/10.1007/978-3-662-03582-5.
R. Lazarsfeld. Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Vol. 48, Springer-Verlag, Berlin (2004).
S. S. Y. Lu and Jörg Winkelmann. Quasiprojective varieties admitting Zariski dense entire holomorphic curves. Forum of Mathematics, (2)24 (2012), 399–418. https://doi.org/10.1515/form.2011.069
M. Nakamaye. Stable base loci of linear series. Mathematische Annalen, (4)318 (2000), 837–847
J. Noguchi. Logarithmic jet spaces and extensions of deFranchis’ theorem. In: Contributions to Several Complex Variables, Aspects of Mathematics, E9, Friedr. Vieweg, Braunschweig (1986), pp. 227–249.
J. Noguchi and J. Winkelmann. Nevanlinna Theory in Several Complex Variables and Diophantine Approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 350, Springer, Tokyo (2014). https://doi.org/10.1007/978-4-431-54571-2.
J. Noguchi, J. Winkelmann, and K. Yamanoi. Degeneracy of holomorphic curves into algebraic varieties. Journal de Mathématiques Pures et Appliquées (9), (3)88 (2007), 293–306. https://doi.org/10.1016/j.matpur.2007.07.003
J. Noguchi, J. Winkelmann, and K. Yamanoi. The second main theorem for holomorphic curves into semi-abelian varieties. II. Forum of Mathematics, (3)20 (2008), 469–503. https://doi.org/10.1515/FORUM.2008.024
J. Noguchi, J. Winkelmann, and K. Yamanoi. Degeneracy of holomorphic curves into algebraic varieties II. Vietnam Journal of Mathematics, (4)41 (2013), 519–525. https://doi.org/10.1007/s10013-013-0051-1
E. Riedl and D. Yang. Application of a grassmannian technique in hypersurfaces. arXiv e-prints (2018). arXiv:1806.02364
E. Rousseau. Hyperbolicité du complémentaire d’une courbe dans \(\mathbb{P}^2\): le cas de deux composantes. Comptes Rendus de l’Académie des Sciences Paris, (8)336 (2003), 635–640. https://doi.org/10.1016/S1631-073X(03)00136-5
E. Rousseau. Logarithmic vector fields and hyperbolicity. The Nagoya Mathema-tical Journal, 195 (2009), 21–40. https://doi.org/10.1017/S0027763000009685
E. Rousseau. Hyperbolicity of geometric orbifolds. Transactions of the American Mathematical Society, (7)362 (2010), 3799–3826. https://doi.org/10.1090/S0002-9947-10-05019-1
X. Roulleau and E. Rousseau. On the hyperbolicity of surfaces of general type with small \(c^2_1\). Journal of the London Mathematical Society (2), (2)87 (2013), 453–477. https://doi.org/10.1112/jlms/jds053
Y.-T. Siu. Hyperbolicity in complex geometry. In: Legacy of Niels Henrik Abel. Springer, Berlin (2004). pp. 543–566
Y.-T. Siu. Hyperbolicity of generic high-degree hypersurfaces in complex projective space. Inventiones Mathematicae, (3)202 (2015), 1069–1166. https://doi.org/10.1007/s00222-015-0584-x
Y.-T. Siu and S.-K. Yeung. Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane. Inventiones Mathematicae, (1–3)124 (1996), 573–618. https://doi.org/10.1007/s002220050064
C. Voisin. On a conjecture of Clemens on rational curves on hypersurfaces. Journal of Differential Geometry, (1)44, 200–213. http://projecteuclid.org/euclid.jdg/1214458743
C. Voisin. A correction: “On a conjecture of Clemens on rational curves on hypersurfaces”. Journal of Differential Geometry, (3)49 (1998), 601–611. http://projecteuclid.org/euclid.jdg/1214461112.
S.-Y. Xie. On the ampleness of the cotangent bundles of complete intersections. Inventiones Mathematicae, (3)212 (2018), 941–996. https://doi.org/10.1007/s00222-017-0783-8
K. Yamanoi. Kobayashi hyperbolicity and higher-dimensional Nevanlinna theory. In: Geometry and Analysis on Manifolds, Progress in Mathematics, Vol. 308. Birkhäuser/Springer, Cham (2015), pp. 209–273.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Brotbek, D., Deng, Y. Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree. Geom. Funct. Anal. 29, 690–750 (2019). https://doi.org/10.1007/s00039-019-00496-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-019-00496-2
Keywords and phrases
- Kobayashi hyperbolicity
- Orbifold hyperbolicity
- Logarithmic-orbifold Kobayashi conjecture
- Second Main Theorem
- Jet differentials
- Logarithmic Demailly tower
- Higher order log connections
- Logarithmic Wronskians