Quasi-projective manifolds uniformized by Carathéodory hyperbolic manifolds and hyperbolicity of their subvarieties

TL;DR

The paper establishes that Carathéodory hyperbolic manifolds admit bounded, real-analytic strictly plurisubharmonic exhaustions, leading to Steinness under strong hyperbolicity and to nondegenerate Bergman metrics on complete Kähler manifolds. It then leverages these analytic tools to connect hyperbolicity properties of universal covers with the algebro-geometric type of subvarieties, proving that compact or quasi-projective varieties with strongly Carathéodory hyperbolic universal covers have subvarieties of general type or log-general type, respectively. A key methodological blend—$L^2$-estimates for $ar ob{d}$, Bergman kernels, Poincaré-type metrics, and Mok’s technique—yields lower bounds on canonical bundle positivity and enables growth of $L^2$ sections that imply big-ness and type. Together, these results illuminate a Lang-type bridge between analytic hyperbolicity and algebraic geometry in both compact and quasi-projective settings, with potential applications to unbounded domains and further Lang conjecture variants.

Abstract

Let $M$ be a Carathéodory hyperbolic complex manifold. We show that $M$ supports a real-analytic bounded strictly plurisubharmonic function. If $M$ is also complete Kähler, we show that $M$ admits the Bergman metric. When $M$ is strongly Carathéodory hyperbolic and is the universal covering of a quasi-projective manifold $X$, the Bergman metric can be estimated in terms of a Poincaré 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.

Theorems & Definitions (35)

• Theorem 1
• Theorem 2
• Conjecture 3: Lang Lang1986
• Theorem 4
• Conjecture 5
• Theorem 6: Main Theorem
• Definition 1.1
• Remark
• Definition 1.2
• Theorem 1.3: = Theorem