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Density conjecture for the typical Hodge locus

Let V\mathbb V be a polarizable Z\mathbb Z-variation of Hodge structure on an irreducible smooth quasi-projective variety SS, with period map Φ:San→Γ\D\Phi:S^{\mathrm{an}}\to\Gamma\backslash D. A special subvariety is typical when its period image meets the corresponding Hodge subvariety with the expected codimension; the typical Hodge locus HL⁡(S,V⊗)typ\operatorname{HL}(S,\mathbb V^{\otimes})_{\mathrm{typ}} is the union of the strict typical special subvarieties of SS for V\mathbb V. Density conjecture for the typical Hodge locus. If

HL⁡(S,V⊗)typ≠∅,\operatorname{HL}(S,\mathbb V^{\otimes})_{\mathrm{typ}}\ne\varnothing,

then it is analytically dense in SS. This predicts that the existence of one strict typical special subvariety forces typical Hodge loci to be analytically dense; the source provides no resolution status, so the conjecture is recorded as open.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Density conjecture for the typical Hodge locus

    Let SS be a parameter space carrying a variation of Hodge structure V\mathbb{V}. Write HL(S,V⊗)typ\textnormal{HL}(S,\mathbb{V}^{\otimes})_{\textnormal{typ}} for the typical Hodge locus.

    Density conjecture. If HL(S,V⊗)typ\textnormal{HL}(S,\mathbb{V}^{\otimes})_{\textnormal{typ}} is non-empty, then it is dense in SS for the analytic topology.

    This conjecture asserts that the existence of one typical Hodge-locus point forces analytic density. It is presented as a conjectural statement in the source, with no resolution indicated.

    source: Gregorio Baldi, Bruno Klingler and Emmanuel Ullmo, “Non-density of the exceptional components of the Noether-Lefschetz locus”, arXiv:2312.11246 (2024).

Sources & referencesView supporting material

Primary source

Gregorio Baldi, Nicholas Miller, Matthew Stover and Emmanuel Ullmo, “Rich representations and superrigidity”, arXiv:2402.03601 (2024).

hodge-locihodge-locushodge-theoryspecial-subvarieties

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