Big Picard theorem for moduli spaces of polarized manifolds

2019/12/24 by Deng, Ya
#32A22 #32Q45 #53C60 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.11442

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→ 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:Δ-\0\→ V from the punctured unit disk to V extends to a holomorphic map of the unit disk Δ 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.

Related

• Picard theorems for moduli spaces of polarized varieties
• On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds
• Hyperbolicity and fundamental groups of complex quasi-projective varieties (I): Maximal quasi-Albanese dimension by Nevanlinna theory
• Hyperbolicity of coarse moduli spaces and isotriviality for certain families
• Holomorphic curves in Base Spaces of Families of Polarized Manifolds