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We introduce the notion of weakly Kähler hyperbolic manifold which generalizes that of Kähler hyperbolic manifold given in the early 1990s 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 Kähler 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.

Weakly Kähler hyperbolic manifolds and the Green–Griffiths–Lang conjecture