On the equidistribution of some Hodge loci

TL;DR

This paper proves the equidistribution of Hodge loci in certain variations of Hodge structure over complex curves, providing asymptotic growth results and implications for elliptic fibrations in K3 surface families.

Contribution

It establishes the equidistribution of Hodge loci for specific non-isotrivial polarized variations of Hodge structure of weight 2 with h^{2,0}=1.

Findings

Hodge loci are equidistributed in the specified setting

Asymptotic growth of Hodge loci under norm conditions

Elliptic fibrations are equidistributed in certain K3 surface families

Abstract

We prove the equidistribution of the Hodge locus for certain non-isotrivial, polarized variations of Hodge structure of weight $2$ with $h^{2,0}=1$ over complex, quasi-projective curves. Given some norm condition, we also give an asymptotic on the growth of the Hodge locus. In particular, this implies the equidistribution of elliptic fibrations in quasi-polarized, non-isotrivial families of $K3$ surfaces.