On the Complexity of Atypical Special Points

Abstract

Given an integral variation of Hodge structure $\mathbb{V}$ on a complex algebraic variety $S$, polarized by some bilinear form $Q:\mathbb{V}\otimes \mathbb{V}\to \mathbb{Z}$, it is believed that the set ${\mathcal{A}}_0^{\text{iso}}\subset S(\mathbb{C})$ of isolated atypical special points associated to $(\mathbb{V},Q)$ forms a finite set. Here we show that the number of such points $s$ is $O(Q(t_s,t_s{)}^{\epsilon })$ for any $\epsilon >0$, where $t_s$ is a minimal integral Hodge tensor defining $s$ (in an appropriate sense). This resolves a conjecture of Grimm and Monnee.