On the Complexity of Atypical Special Points

2025/12/04 by Urbanik, David
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.04491

Abstract

Given an integral variation of Hodge structure \mathbbV on a complex algebraic variety S, polarized by some bilinear form Q : \mathbbV ⊗ \mathbbV → ℤ, it is believed that the set A^\textrmiso₀ ⊂ S(ℂ) of isolated atypical special points associated to (\mathbbV, Q) forms a finite set. Here we show that the number of such points s is O(Q(t_s, t_s)^ε) for any ε > 0, where t_s is a minimal integral Hodge tensor defining s (in an appropriate sense). This resolves a conjecture of Grimm and Monnee.

Citations

• Degrees of Hodge Loci
• Geometric G-functions and Atypicality
• On the distribution of the Hodge locus
• Tame topology of arithmetic quotients and algebraicity of Hodge loci
• The Ax-Schanuel conjecture for variations of Hodge structures
• Hodge loci and atypical intersections: conjectures
• Fields of definition of Hodge loci
• The primitive cohomology lattice of a complete intersection
• On the Locus of Hodge Classes

Related

• On the distribution of the Hodge locus
• Hodge loci and atypical intersections: conjectures
• On the geometric André-Oort conjecture for variations of Hodge structures
• The Ax-Schanuel conjecture for variations of Hodge structures
• Tame topology of arithmetic quotients and algebraicity of Hodge loci