Algebraically hyperbolic manifolds have finite automorphism groups

TL;DR

This paper proves that algebraically hyperbolic projective manifolds, which generalize Kobayashi hyperbolic manifolds, have finite automorphism groups, extending known results in complex geometry.

Contribution

It establishes that algebraic hyperbolicity implies finiteness of automorphism groups for projective manifolds, broadening the class of manifolds with this property.

Findings

Algebraically hyperbolic manifolds have finite automorphism groups.

Extension of Kobayashi hyperbolicity results to algebraic hyperbolicity.

Provides new insights into the symmetry properties of hyperbolic manifolds.

Abstract

A projective manifold $M$ is algebraically hyperbolic if there exists a positive constant $A$ such that the degree of any curve of genus $g$ on $M$ is bounded from above by $A(g-1)$. A classical result is that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. Here we prove that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups.