Non-Degeneracy of Kobayashi Volume Measures for Singular Directed Varieties

TL;DR

This paper proves that for singular directed varieties with a big canonical sheaf, the Kobayashi volume measure is generically hyperbolic, extending hyperbolicity results to singular settings.

Contribution

It establishes the hyperbolicity of Kobayashi volume measures for singular directed varieties when the canonical sheaf is big, a new result in complex geometry.

Findings

Kobayashi volume measure is hyperbolic for generic singular directed varieties

Hyperbolicity holds when the canonical sheaf is big

Extends hyperbolicity results to singular varieties

Abstract

In this note, we prove the generic Kobayashi volume measure hyperbolicity of singular directed varieties $(X, V)$, as soon as the canonical sheaf $\mathcal{K}\_V$ of $V$ is big in the sense of Demailly.