Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree

TL;DR

This paper proves that the complements of high-degree general hypersurfaces in projective manifolds are Kobayashi hyperbolic, confirming a longstanding conjecture and providing explicit degree bounds using jet differential techniques.

Contribution

It establishes Kobayashi hyperbolicity for complements of general hypersurfaces of large degree, introduces effective degree bounds, and extends results to orbifold hyperbolicity and cyclic covers.

Findings

Proves hyperbolicity of hypersurface complements with explicit degree bounds

Constructs logarithmic Wronskians and jet differentials for hyperbolicity

Extends hyperbolicity results to orbifold and cyclic cover cases

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.