Weakly K\"ahler hyperbolic manifolds and the Green--Griffiths--Lang conjecture

TL;DR

This paper introduces weakly Kähler hyperbolic manifolds, explores their spectral properties, and demonstrates their significance in complex geometry by confirming aspects of the Green--Griffiths--Lang conjecture.

Contribution

It generalizes the concept of Kähler hyperbolic manifolds, establishes their spectral gap, and applies these results to verify key conjectures in complex geometry.

Findings

Weakly Kähler hyperbolic manifolds are of general type.

Spectral gap results are established for these manifolds.

The study confirms various aspects of the Green--Griffiths--Lang conjecture for this class.

Abstract

We introduce the notion of weakly K\"ahler hyperbolic manifold which generalizes that of K\"ahler hyperbolic manifold given in the early '90s by M. Gromov, and establish its basic features. We then investigate its spectral properties and show a spectral gap result (on a suitable modification). As applications, we prove that weakly K\"ahler hyperbolic manifolds are of general type and we study the geometry of their subvarieties and entire curves, verifying -- among other things -- various aspects of the Lang and the Green--Griffiths conjectures for this class of manifolds.