  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.
