Quasi-projective manifolds uniformized by Carath\'eodory hyperbolic manifolds and hyperbolicity of their subvarieties

TL;DR

The paper investigates the geometric properties of Carathéodory hyperbolic manifolds and their subvarieties, establishing conditions for the existence of special metrics and confirming conjectures about their algebraic types.

Contribution

It proves the existence of bounded strictly plurisubharmonic functions and Bergman metrics on Carathéodory hyperbolic manifolds, and shows subvarieties are of log-general or general type, supporting Lang's conjecture.

Findings

Existence of a real-analytic bounded strictly plurisubharmonic function on M.

Bergman metric exists if M is complete Kähler.

Subvarieties are of log-general or general type.

Abstract

Let $M$ be a Carath\'eodory hyperbolic complex manifold. We show that $M$ supports a real-analytic bounded strictly plurisubharmonic function. If $M$ is also complete K\"ahler, we show that $M$ admits the Bergman metric. When $M$ is strongly Carath\'eodory hyperbolic and is the universal covering of a quasi-projective manifold $X$, the Bergman metric can be estimated in terms of a Poincar\'e type metric on $X$. It is also proved that any quasi-projective (resp. projective) subvariety of $X$ is of log-general type (resp. general type), a result consistent with a conjecture of Lang.