On the fields of definition of Hodge loci

TL;DR

This paper investigates the algebraic fields over which Hodge loci are defined, proving a conjecture for certain cases and reducing the general conjecture to the case of special points, advancing understanding of Hodge structures.

Contribution

The paper proves the conjecture for special subvarieties with a simple monodromy condition and reduces the general conjecture to the case of special points.

Findings

Proved the conjecture for subvarieties with simple monodromy.

Reduced the conjecture to the case of special points.

Established that special subvarieties are defined over algebraic closures of number fields.

Abstract

A polarizable variation of Hodge structure over a smooth complex quasi projective variety $S$ is said to be defined over a number field $L$ if $S$ and the algebraic connection associated to the variation are both defined over $L$. Conjecturally any special subvariety (also called "an irreducible component of the Hodge locus) for such variations is defined over $\overline{\mathbb{Q}}$, and its Galois conjugates are also special subvarieties. We prove this conjecture for special subvarieties satisfying a simple monodromy condition. As a corollary we reduce the conjecture that special subvarieties for variation of Hodge structures defined over a number field are defined over $\overline{\mathbb{Q}}$ to the case of special points.