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.

中文翻译：

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小林高双曲面补全的双曲性

在本文中，我们证明了在任何射影流形中，足够大程度的一般超曲面的补都是小林双曲。我们还提供了有效的学位下限。这证实了S. Kobayashi在1970年所做的一个猜想。基于射流微分理论，我们的证明是通过将问题简化为具有强双曲性质的特定示例来获得的。这种方法依赖于高阶对数连接的构建，这使我们能够构建对数Wronskians。这些对数Wronskians是我们能够构建的更一般的对数射流差动的基础。作为我们证明的副产品，我们证明了Campana意义上的一般几何球面的球面双曲率的更一般的结果，仅具有一个组成部分，并且具有很大的重复性。我们还建立了与一般超曲面相交的全纯全曲线的第二主定理类型结果，我们证明了一般超曲面的循环覆盖的Kobayashi双曲性，同样在所有这些超曲面的度上都具有明显的下限。