Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures

Abstract

In this paper, we study various hyperbolicity properties for a quasi-compact Kähler manifold $U$ which admits a complex polarized variation of Hodge structures so that each fiber of the period map is zero-dimensional. In the first part, we prove that $U$ is algebraically hyperbolic and that the generalized big Picard theorem holds for $U$. In the second part, we prove that there is a finite étale cover $\tilde{U}$ of $U$ from a quasi-projective manifold $\tilde{U}$ such that any projective compactification $X$ of $\tilde{U}$ is Picard hyperbolic modulo the boundary $X-\tilde{U}$, and any irreducible subvariety of $X$ not contained in $X-\tilde{U}$ is of general type. This result coarsely incorporates previous works by Nadel, Rousseau, Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients of bounded symmetric domains by torsion-free lattices.

31 pages. Final version, to appear in Épijournal de Géométrie Algébrique

References in corpus (8)

• Shimura Varieties and Moduli
• Semi-positivity from Higgs bundles
• Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications
• Increasing hyperbolicity of varieties supporting a variation of Hodge structures with level structures
• Picard theorems for moduli spaces of polarized varieties
• Hyperbolicity and fundamental groups of complex quasi-projective varieties
• Picard hyperbolicity of manifolds admitting nilpotent harmonic bundles
• Integral points on algebraic subvarieties of period domains: from number fields to finitely generated fields

Cited by in corpus (1)

• Carathéodory hyperbolicity, volume estimates and level structures over function fields