On the Complexity of Atypical Special Points
David Urbanik
TL;DR
The paper addresses finiteness questions for isolated atypical special points arising from a polarized integral variation of Hodge structure on a complex variety. It introduces a precise complexity bound: the count of such points defined by tensors of bounded rank and height grows subpolynomially, $O(q^{\varepsilon})$ for any $\varepsilon>0$. The approach combines period-domain definability, height bounds, and Ax–Schanuel type results within a Pila–Wilkie counting framework, reducing to zero-dimensional period domains and exploiting the $\mathbb{Q}$-simplicity of the adjoint monodromy group. As a key consequence, the Grimm–Monnee conjecture is resolved in the case where $\mathbf{G}_S^{\mathrm{ad}}$ is $\mathbb{Q}$-simple, providing a significant step in understanding unlikely intersections in Hodge theory and Zilber–Pink style problems.
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.