On the closure of the Hodge locus of positive period dimension

Klingler, Bruno; Otwinowska, Anna

doi:10.1007/s00222-021-01042-4

Cited by 9 publications

(14 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The distribution of the Hodge locus has been investigated independently and concomitantly by Baldi, Klingler, and Ullmo in [BKU21]. In addition to striking results on the atypical Hodge locus (see also [KO21] for prior work), they prove several properties about the typical Hodge locus that echo the present work, namely that the typical Hodge locus is either empty or dense, and is always empty when the level is at least three.…”

Section: Introductionsupporting

confidence: 71%

See 1 more Smart Citation

Equidistribution of Hodge loci II

Tayou¹,

Tholozan²

2023

Compositio Math.

View full text Add to dashboard Cite

Let $\mathbb {V}$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$ . In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, the pull–push form. In particular, it is always analytically dense when the pull–push form does not vanish. When the weight is two, the Hodge numbers are $(q,p,q)$ and the dimension of $S$ is least $rq$ , we prove that the typical locus where the Picard rank is at least $r$ is equidistributed in $S$ with respect to the volume form $c_q^r$ , where $c_q$ is the $q$ th Chern form of the Hodge bundle. We obtain also several equidistribution results of the typical locus in Shimura varieties: a criterion for the density of the typical Hodge loci of a variety in $\mathcal {A}_g$ , equidistribution of certain families of CM points and equidistribution of Hecke translates of curves and surfaces in $\mathcal {A}_g$ . These results are proved in the much broader context of dynamics on homogeneous spaces of Lie groups which are of independent interest. The pull–push form appears in this greater generality, we provide several tools to determine it, and we compute it in many examples.

show abstract

Section: Introductionsupporting

confidence: 71%

“…Theorem 1.2 and Proposition 1.5 have also been independently studied by Baldi, Klingler, and Ullmo [BKU21], see also the prior work of Klingler and Otwinowska [KO21]. Moreover, the authors proved in [BKU21, Theorem 2.3] that the condition in Proposition 1.5 is always satisfied whenever has level more than three.…”

Section: Introductionmentioning

confidence: 92%

Equidistribution of Hodge loci II

Tayou¹,

Tholozan²

2023

Compositio Math.

View full text Add to dashboard Cite

show abstract

“…Then one sees that the above loci are also equal to W S pos . The fact that HL pos = W S pos is already used crucially in [KO21]. Theorem 3.9 demonstrates that we have good control over the arithmetic of special subvarieties of positive period dimension (at least when G der S is simple), whereas we cannot say anything about special points or ℓ-Galois special points.…”

Section: Weakly Special Subvarietiesmentioning

confidence: 99%

$\ell$-Galois special subvarieties and the Mumford-Tate conjecture

Kreutz¹

2021

Preprint

View full text Add to dashboard Cite

We introduce ℓ-Galois special subvarieties as an ℓ-adic analogue of the Hodgetheoretic notion of a special subvariety. The Mumford-Tate conjecture predicts that both notions are equivalent. We prove this equivalence for subvarieties satisfying a simple monodromy condition. As an application we see that if the derived group of the generic Mumford-Tate group of a family is simple, then the positive part of the Hodge locus coincides with the positive part of the ℓ-Galois exceptional locus. Finally, we show that the Mumford-Tate conjecture for abelian varieties is equivalent to a conjecture about the existence of positive dimensional algebraic subvarieties in Shimura varieties with certain Galois theoretic properties.

show abstract

“…It is known that the algebraic monodromy group of a variety is always a -normal subgroup of its Mumford–Tate group by a result of André and Deligne [Yve92], and that any special subvariety of is in fact weakly special [KO21, Definition 3.1]. Thus, studying the set of weakly special subvarieties is a generalization of the problem of studying the set of special subvarieties.…”

Section: Introductionmentioning

confidence: 99%

“…Here we really mean positive-dimensional in the Hodge-theoretic sense of[KO21, Definition 1.3], which coincides with being positive-dimensional as a subvariety in the quasi-finite period map case.3 This is due to the simplicity of derived subgroup of the Mumford-Tate group of the variation, this latter fact being a consequence of the result that the global algebraic monodromy group is maximal due to[Bea86].…”

mentioning

confidence: 99%

Sets of special subvarieties of bounded degree

Urbanik¹

2023

Compositio Math.

View full text Add to dashboard Cite

Let $f : X \to S$ be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base $S$ , and let $\mathbb {V} = R^{2k} f_{*} \mathbb {Z}(k)$ be the integral variation of Hodge structure coming from degree $2k$ cohomology it induces. Associated to $\mathbb {V}$ one has the so-called Hodge locus $\textrm {HL}(S) \subset S$ , which is a countable union of ‘special’ algebraic subvarieties of $S$ parametrizing those fibres of $\mathbb {V}$ possessing extra Hodge tensors (and so, conjecturally, those fibres of $f$ possessing extra algebraic cycles). The special subvarieties belong to a larger class of so-called weakly special subvarieties, which are subvarieties of $S$ maximal for their algebraic monodromy groups. For each positive integer $d$ , we give an algorithm to compute the set of all weakly special subvarieties $Z \subset S$ of degree at most $d$ (with the degree taken relative to a choice of projective compactification $S \subset \overline {S}$ and very ample line bundle $\mathcal {L}$ on $\overline {S}$ ). As a corollary of our algorithm we prove conjectures of Daw–Ren and Daw–Javanpeykar–Kühne on the finiteness of sets of special and weakly special subvarieties of bounded degree.

show abstract

On the closure of the Hodge locus of positive period dimension

Cited by 9 publications

References 22 publications

Equidistribution of Hodge loci II

Equidistribution of Hodge loci II

$\ell$-Galois special subvarieties and the Mumford-Tate conjecture

Sets of special subvarieties of bounded degree

Contact Info

Product

Resources

About