Complex-analytic intermediate hyperbolicity, and finiteness properties

TL;DR

This paper explores the finiteness of automorphism groups in complex manifolds with intermediate hyperbolicity properties, linking hyperbolic conditions to algebraic and geometric finiteness results.

Contribution

It establishes finiteness of automorphism groups for (dim(X)-1)-analytically hyperbolic manifolds and introduces intermediate Picard hyperbolicity.

Findings

Finite automorphism groups for (dim(X)-1)-analytically hyperbolic manifolds

Extension of finiteness results to pseudo-hyperbolic projective manifolds under certain conditions

Introduction of intermediate Picard hyperbolicity concept

Abstract

Motivated by the finiteness of the set of automorphisms Aut(X) of a projective manifold X, and by Kobayashi-Ochiai's conjecture that a projective manifold dim(X)-analytically hyperbolic (also known as strongly measure hyperbolic) should be of general type, we investigate the finiteness properties of Aut(X) for a complex manifold satisfying a (pseudo-) intermediate hyperbolicity property. We first show that a complex manifold X which is (dim(X) -- 1)-analytically hyperbolic has indeed finite automorphisms group. We then obtain a similar statement for a pseudo-(dim(X) -- 1)-analytically hyperbolic, strongly measure hyperbolic projective manifold X, 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.