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In this paper, we prove that in any projective manifold, the complements ofgeneral hypersurfaces of sufficiently large degree are Kobayashi hyperbolic. Wealso provide an effective lower bound on the degree. This confirms a conjectureby 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 examplewith strong hyperbolicity properties. This approach relies the construction ofhigher order logarithmic connections allowing us to construct logarithmicWronskians. These logarithmic Wronskians are the building blocks of the moregeneral logarithmic jet differentials we are able to construct. As a byproductof our proof, we prove a more general result on the orbifold hyperbolicity forgeneric geometric orbifolds in the sense of Campana, with only one componentand large multiplicities. We also establish a Second Main theorem type resultfor holomorphic entire curves intersecting general hypersurfaces, and we provethe Kobayashi hyperbolicity of the cyclic cover of a general hypersurface,again with an explicit lower bound on the degree of all these hypersurfaces.

Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree