Algebraic Hodge generic points are dense

Summary · qwen2.5:32b

The study proves that algebraic Hodge generic points are dense within complex varieties over $\overline{\mathbb{Q}}$, contributing to the Grothendieck period conjecture under a large monodromy condition, and it establishes new cases of the Mumford-Tate conjecture. A key technical achievement is a novel result on relations among solutions of $G$-operators, utilizing height estimates by Bombieri and André.

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Article discusses Algebraic Hodge theory and proves density of certain points in complex algebraic geometry, related to the Grothendieck period conjecture and Mumford-Tate conjecture.

Excerpt

arXiv:2606.08882v2 Announce Type: replace Abstract: Let $f: X \to S$ be a quasi-projective family of varieties defined over $\overline{\mathbb{Q}} \subset \mathbb{C}$. We show that the points of $S(\overline{\mathbb{Q}})$ that are Hodge generic for the variation of Hodge structures associated to $f$ are analytically dense in $S(\mathbb{C})$. In fact, in the spirit of the Grothendieck period conjecture and under a large monodromy assumption, we prove the density of the points of $S(\overline{\mathbb{Q}})$ where the periods of the fibre do not satisfy extra relations 'up to degree $\delta$'. As a by-product, we also establish new instances of the Mumford-Tate conjecture, beyond the realm of abelian motives. When the base $S$ is a curve, we provide quantitative estimates for points satisfying these properties. The main technical contribution is a new result on relations satisfied by solutions of $G$-operators, which relies on height estimates due to Bombieri and Andr\'e.