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 $\mathcal{A}^{\textrm{iso}}_{0} \subset S(\mathbb{C})$ 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 $O(Q(t_{s}, t_{s})^{\varepsilon})$ for any $\varepsilon > 0$, where $t_{s}$ is a minimal integral Hodge tensor defining $s$ (in an appropriate sense). This resolves a conjecture of Grimm and Monnee.