Uploaded June 2026 | Updated September 2026, 3 weeks ago

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Special Year Learning Seminar

Topic: Algebraic Hodge Generic Points are Dense
Speaker: Gregorio Baldi
Affiliation: Institute for Advanced Study
Date: June 12, 2026
Simonyi 101

Let f:X→S
be a quasi-projective family of varieties defined over ℚ⎯⎯⎯⎯⎯⊂ℂ
. We show that the points of S(ℚ⎯⎯⎯⎯⎯)
that are Hodge generic for the variation of Hodge structures associated to f
are analytically dense in S(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(ℚ⎯⎯⎯⎯⎯)
where the periods of the fibre do not satisfy extra relations ``up to degree δ
''. 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.

Joint work with G. Binyamini and D. Urbanik.