On the hyperbolicity of general hypersurfaces

TL;DR

This paper proves that sufficiently ample general hypersurfaces in smooth projective varieties are hyperbolic by constructing special hypersurfaces with properties linked to jet space geometry, advancing understanding of Kobayashi's conjecture.

Contribution

It introduces a new Wronskian construction relating jet differentials to global sections, enabling the proof of hyperbolicity for a broad class of hypersurfaces.

Findings

Proves hyperbolicity of sufficiently ample general hypersurfaces.

Introduces a Wronskian construction for jet differentials.

Establishes a Zariski open property implying hyperbolicity.

Abstract

In 1970, Kobayashi conjectured that general hypersurfaces of sufficiently large degree in $P^n$ are hyperbolic. In this paper we prove that a general sufficiently ample hypersurface in a smooth projective variety is hyperbolic. To prove this statement, we construct hypersurfaces satisfying a property which is Zariski open and which implies hyperbolicity. These hypersurfaces are chosen such that the geometry of their higher order jet spaces can be related to the geometry of a universal family of complete intersections. To do so, we introduce a Wronskian construction which associates a (twisted) jet differential to every finite family of global sections of a line bundle.