On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds

Abstract

We show that the moduli stack $\mathscr {M}\sb h$ of canonically polarized complex manifolds with Hilbert polynomial $h$ is Brody hyperbolic. Hence if $M\sb h$ denotes the corresponding coarse moduli scheme, and if $U \to M\sb h$ is a quasi-finite morphism, induced by a family, then there are no nonconstant holomorphic maps $\mathbb {C}\to U$.