On the distribution of mixed Hodge loci

TL;DR

This paper investigates the distribution and density of Hodge loci in variations of mixed Hodge structures, establishing conditions under which these loci are dense, empty, or Zariski-closed, and classifying cases of dense transverse Hodge loci.

Contribution

It provides new criteria for the density and emptiness of Hodge loci, classifies variations with dense transverse Hodge loci, and refines classical results in the context of mixed Hodge structures.

Findings

If the typical Hodge locus is non-empty, the full Hodge locus is dense in the Zariski topology.

Large monodromy and high level imply the typical Hodge locus is empty and the full locus is a strict Zariski-closed subset.

Non-emptiness of the transverse Hodge locus is equivalent to its Zariski-density in the base variety.

Abstract

Let $\mathbb{V}$ be an admissible and graded-polarized integral variation of mixed Hodge structures over a smooth and irreducible complex algebraic variety $S$. We show that if the typical Hodge locus $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{typ}$ of $\mathbb{V}$ is non-empty, the full Hodge locus $\mathrm{HL}(S,\mathbb{V}^\otimes)$ is dense in $S$ for the Zariski topology. In an other direction, we show that if the associated graded variation $\mathrm{Gr}(\mathbb{V})$ for the weight filtration has large monodromy and level at least 3 in the sense of Baldi- Klingler-Ullmo, the typical Hodge locus of $\mathbb{V}$ is empty, and the full Hodge locus of $\mathbb{V}$ is a strict Zariski-closed subset of $S$, at least if one restricts to its factorwise positive dimensional part, improving a classical result of Brosnan-Pearlstein-Schnell in this situation. These results follow from a detailed study of the transverse part $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ of the Hodge locus of $S$ for $\mathbb{V}$, a subset which contains $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{typ}$ and whose Zariski-density in $S$ is equivalent, under the Zilber-Pink conjecture for $\mathbb{V}$, to the Zariski-density of $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{typ}$. We show that non-emptiness of $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ is equivalent to its Zariski-density in $S$, we completely classify variations whose transverse Hodge locus $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ is Zariski-dense, and we prove an independent criterion ensuring that $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ is empty.