Intermediate hyperbolicity of varieties supporting a variation of Hodge structure

Abstract

Let $\bar{V}$ be a connected smooth complex projective variety. Let $D \subset \bar{V}$ be a normal crossing divisor. Let $\mathbb{V}$ be a complex polarizable variation of Hodge structure on $V := \bar{V} \setminus D$. Suppose that the period map of $\mathbb{V}$ is immersive at a point of $V$. We prove that for every integer $p$ with $1 \leq p \leq \dim V$, the vector bundle $\Omega_{\bar{V}}^p(\log D)$ is L-big (i.e. the tautological line bundle on $\mathbb{P}\Omega_{\bar{V}}^p(\log D)$ is big). If the local monodromy is quasi-unipotent, we give a method to determine an $m \in \mathbb{N}$ such that if $p > m$, then $\Omega_{\bar{V}}^p(\log D)$ is moreover Viehweg-big. We give the optimal value of $m$ explicitly when $V$ is a locally symmetric variety. We prove that if $V$ is a finite \'etale cover of the fine moduli space of smooth quintic threefolds, the result holds for $m = 90$ (in this case $\dim V = 101$). The proof of the previous results is based on a study of an augmented base locus associated with $\Omega_{\bar{V}}^p(\log D)$. In the case of locally symmetric varieties, we introduce ``higher degree characteristic subvarieties'', generalizing the characteristic subvariety defined by Mok in the case $p = 1$, and prove that they coincide with these augmented base loci.