Existence and density of typical Hodge loci

Khelifa, Nazim; Urbanik, David

doi:10.48550/arxiv.2303.16179

Cited by 1 publication

(2 citation statements)

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“…asserts that we would expect (non-empty) components of NL L of codimension h 2,0 ev . This is exactly the "admissible" condition needed to apply the main results of [15,23], which therefore establishes the existence of components of the desired codimension. In op.…”

Section: Zilber-pink and The Noether-lefschetz Locus For Arbitrary Th...mentioning

confidence: 80%

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On the distribution of the Hodge locus

Baldi,

Klingler,

Ullmo

2023

Invent. math.

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Given a polarizable ℤ-variation of Hodge structures $\mathbb{V}$ V over a complex smooth quasi-projective base $S$ S , a classical result of Cattani, Deligne and Kaplan says that its Hodge locus (i.e. the locus where exceptional Hodge tensors appear) is a countable union of irreducible algebraic subvarieties of $S$ S , called the special subvarieties for $\mathbb{V}$ V . Our main result in this paper is that, if the level of ${\mathbb{V}}$ V is at least 3, this Hodge locus is in fact a finite union of such special subvarieties (hence is algebraic), at least if we restrict ourselves to the Hodge locus factorwise of positive period dimension (Theorem 1.5). For instance the Hodge locus of positive period dimension of the universal family of degree $d$ d smooth hypersurfaces in $\mathbf{P}^{n+1}_{\mathbb{C}}$ P C n + 1 , $n\geq 3$ n ≥ 3 , $d\geq 5$ d ≥ 5 and $(n,d)\neq (4,5)$ ( n , d ) ≠ ( 4 , 5 ) , is algebraic. On the other hand we prove that in level 1 or 2, the Hodge locus is analytically dense in $S^{\mbox{an}}$ S an as soon as it contains one typical special subvariety. These results follow from a complete elucidation of the distribution in $S$ S of the special subvarieties in terms of typical/atypical intersections, with the exception of the atypical special subvarieties of zero period dimension.

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Section: Zilber-pink and The Noether-lefschetz Locus For Arbitrary Th...mentioning

confidence: 80%

“…See also [10] for the construction of some explicit exceptional components. We refer also to the recent works [15,23] (inspired by the Zilber-Pink paradigm proposed in [1]). In particular such results show, abstractly, the existence of general components in NL d and the density of their union.…”

Section: Introductionmentioning

confidence: 99%