Big Picard theorem for moduli spaces of polarized manifolds

Abstract

Consider a smooth projective family of complex polarized manifolds with semi-ample canonical sheaf over a quasi-projective manifold $V$. When the associated moduli map $V\to P_h$ from the base to coarse moduli space is quasi-finite, we prove that the generalized big Picard theorem holds for the base manifold $V$: for any projective compactification $Y$ of $V$, any holomorphic map $f:\Delta-\{0\}\to V$ from the punctured unit disk to $V$ extends to a holomorphic map of the unit disk $\Delta$ into $Y$. This result generalizes our previous work on the Brody hyperbolicity of $V$ (i.e. there are no entire curves on $V$), as well as a more recent work by Lu-Sun-Zuo on the Borel hyperbolicity of $V$ (i.e. any holomorphic map from a quasi-projective variety to $V$ is algebraic). We also obtain generalized big Picard theorem for bases of log Calabi-Yau families.