Fields of definition of Hodge loci

TL;DR

This paper proves that irreducible components of Hodge loci are defined over algebraically closed subfields under certain conditions, linking to the theory of absolute Hodge classes and using spread techniques.

Contribution

It establishes that Hodge loci components are defined over algebraically closed subfields when the base variety and Hodge bundle are defined over such fields, extending previous understanding.

Findings

Hodge locus components are defined over algebraically closed subfields.

The result applies to Hodge loci inside Hodge bundles with connections.

The proof uses the spread of Hodge loci and parallels the zero locus of admissible normal functions.

Abstract

We show that an irreducible component of the Hodge locus of a polarizable variation of Hodge structure of weight 0 on a smooth complex variety X is defined over an algebraically closed subfield k of finite transcendence degree if X is defined over k and the component contains a k-rational point. We also prove a similar assertion for the Hodge locus inside the Hodge bundle if the Hodge bundle together with the connection is defined over k. This is closely related with the theory of absolute Hodge classes. The proof uses the spread of the Hodge locus, and is quite similar to the case of the zero locus of an admissible normal function.