On the closure of the positive Hodge locus

TL;DR

This paper investigates the structure of the positive Hodge locus in variations of Hodge structures, establishing conditions under which its union of components is either algebraic or Zariski-dense.

Contribution

It proves a dichotomy for the union of positive components of the Hodge locus in certain variations, advancing understanding of their algebraic and density properties.

Findings

Union of positive Hodge locus components is either algebraic or Zariski-dense.

The result applies to variations with non-product generic Mumford-Tate group.

Provides a classification of the closure properties of the positive Hodge locus.

Abstract

Given a variation of Hodge structures on a quasi-projective base $S$, whose generic Mumford-Tate group is non-product, we prove that the (countable) union of positive components of the Hodge locus is either an algebraic subvariety of $S$, or is Zariski-dense in $S$.