Complex-analytic intermediate hyperbolicity, and finiteness properties

Abstract

Motivated by the finiteness of the set of automorphisms [Formula: see text] of a projective manifold of general type [Formula: see text], and by Kobayashi–Ochiai’s conjecture that a projective manifold [Formula: see text]-analytically hyperbolic (also known as strongly measure hyperbolic) should be of general type, we investigate the finiteness properties of [Formula: see text] for a complex manifold satisfying a (pseudo-) intermediate hyperbolicity property. We first show that a complex manifold [Formula: see text] which is [Formula: see text]-analytically hyperbolic has indeed finite automorphisms group. We then obtain a similar statement for a pseudo-[Formula: see text]-analytically hyperbolic, strongly measure hyperbolic projective manifold [Formula: see text], under an additional hypothesis on the size of the degeneracy set. Some of the properties used during the proofs lead us to introduce a notion of intermediate Picard hyperbolicity, which we last discuss.