Properties of Caratheodory measure hyperbolic universal covers of compact Kahler manifolds

TL;DR

This paper investigates properties of universal covers of compact Kahler manifolds under Caratheodory measure hyperbolicity, establishing inequalities that relate their volume forms and solving a related conjecture.

Contribution

It introduces new inequalities linking the canonical bundle volume and Caratheodory measure of universal covers, addressing a previously unresolved conjecture.

Findings

Established an inequality between the volume of the canonical bundle and Caratheodory measure.

Proved an inequality involving restricted volume and restricted Caratheodory measure.

Solved a conjecture related to these inequalities in the context of compact Kahler manifolds.

Abstract

This article explores some properties of universal covers of compact Kahler manifolds, under the assumption of Caratheodory measure hyperbolicity. In particular, by comparing invariant volume forms, an inequality is established between the volume of canonical bundle of a compact Kahler manifolds and the Caratheodory measure of its universal cover (similar result as in [Kikuta 10]). Using similar method, an inequality is established between the restricted volume of canonical bundle of a compact Kahler manifolds and the restricted Caratheodory measure of its covering, solving a conjecture in [Kikuta 13].