Geometric and Functional Analysis

, Volume 29, Issue 3, pp 690–750 | Cite as

Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree

  • Damian Brotbek
  • Ya DengEmail author


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.

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 

List of Symbols

\(f:({{\mathbb {C}}},0)\rightarrow X\)

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


Directed manifolds


Demailly–Semple k-jet tower of the directed manifold (XV)


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


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


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 (XD) (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)

\(\omega _{D}^{\prime }(\bullet )\)

Defined in (2.12) or (2.18)

\( {\mathfrak {w}}_{{X}_k(D)}\)

k-th logarithmic Wronskian ideal sheaf of the log manifold (XD)

\(\mathbb {L}\)

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 \((Y,\Delta )\).

Mathematics Subject Classification

32Q45 30D35 14E99 



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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Centre de Mathématiques Laurent SchwartzÉcole polytechniquePalaiseau CEDEXFrance
  2. 2.Institut de Recherche Mathématique AvancéeUniversité de StrasbourgStrasbourg CEDEXFrance

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