Orbifold hyperbolicity

TL;DR

This paper develops the theory of jet bundles in the orbifold setting, revealing new phenomena and establishing hyperbolicity results for certain orbifold surfaces, while contrasting with classical results for smooth varieties.

Contribution

It introduces and studies orbifold jet bundles, showing limitations of existing methods and providing new hyperbolicity results for orbifold surfaces.

Findings

Orbifold jet bundles exhibit different properties than classical jet bundles.

Some orbifold pairs of general type do not admit global jet differentials.

The paper proves hyperbolicity for certain orbifold surfaces.

Abstract

We define and study jet bundles in the geometric orbifold category. We show that the usual arguments from the compact and the logarithmic settings do not all extend to this more general framework. This is illustrated by simple examples of orbifold pairs of general type that do not admit any global jet differential, even if some of these examples satisfy the Green-Griffiths-Lang conjecture. This contrasts with an important result of Demailly (2010) proving that compact varieties of general type always admit jet differentials. We illustrate the usefulness of the study of orbifold jets by establishing the hyperbolicity of some orbifold surfaces, that cannot be derived from the current techniques in Nevanlinna's theory. We also conjecture that Demailly's theorem should hold for orbifold pairs with smooth boundary divisors under a certain natural multiplicity condition, and provide some evidence towards it.