Finsler metrics and Kobayashi hyperbolicity of the moduli spaces of canonically polarized manifolds

Abstract

We show that the base complex manifold of an effectively parametrized holomorphic family of compact canonically polarized complex manifolds admits a smooth invariant Finsler metric whose holomorphic sectional curvature is bounded above by a negative constant. As a consequence, we show that such a base manifold is Kobayashi hyperbolic.