On the Complexity of Atypical Special Points

TL;DR

This paper proves a bound on the number of isolated atypical special points in a complex algebraic variety, confirming a conjecture and showing their finiteness with a specific growth rate related to Hodge tensors.

Contribution

It establishes an upper bound on the number of isolated atypical special points, resolving a conjecture by Grimm and Monnee.

Findings

Number of atypical special points is bounded by a polynomial in the Hodge tensor norm.

The set of such points is finite under the given conditions.

The result confirms the finiteness conjecture for these 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.