Effective estimates for the degrees of maximal special subvarieties

Daw, Christopher; Javanpeykar, Ariyan; Kühne, Lars

doi:10.1007/s00029-019-0528-1

Cited by 3 publications

(5 citation statements)

References 21 publications

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“…Effective results on the André-Oort and the Zilber-Pink conjectures are still relatively sparse. This work improves upon a previous work of the second author with Javanpeykar and Kühne [7] which, by entirely different methods, gave effective degree bounds for so-called non-facteur maximal special subvarieties. We refer to the introduction of the latter for references to several earlier works.…”

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confidence: 74%

Effective computations for weakly optimal subvarieties

Binyamini,

Daw

2021

Preprint

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Ren and the second author established that the weakly optimal subvarieties (e.g. maximal weakly special subvarieties) of a subvariety V of a Shimura variety arise in finitely many families. In this article, we refine this theorem by (1) constructing a finite collection of algebraic families whose fibers are precisely the weakly optimal subvarieties of V ; (2) obtaining effective degree bounds on the weakly optimal locus and its individual members; (3) describing an effective procedure to determine the weakly optimal locus.

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confidence: 74%

Effective computations for weakly optimal subvarieties

Binyamini,

Daw

2021

Preprint

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“…As explained in [DR18, § 10] and [DJK20, Remark 3.8], Corollary 6.2 also resolves [DR18, Conjecture 10.3] of Daw and Ren and the conjecture in [DJK20] of Daw, Javanpekkar and Kühne that there should be finitely many non-factor special subvarieties of bounded degree.…”

Section: Applicationssupporting

confidence: 63%

“…In a paper [DJK20] by Daw, Javanpeykar and Kühne, a related conjecture is made in Remark 3.8, which is a special case of the following statement.…”

Section: Urbanikmentioning

confidence: 86%

Sets of special subvarieties of bounded degree

Urbanik¹

2023

Compositio Math.

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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.

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“…3.8], Corollary 6.2 also resolves [DR18, Conj. 10.3] of Daw-Ren and the conjecture in [DJK20] of Daw-Javanpekkar-Kühne that there should be finitely many non-facteur special subvarieties of bounded degree.…”

Section: Resultsmentioning

confidence: 99%

“…In a paper [DJK20] by Daw-Javanpeykar-Kühne a related conjecture is made in Remark 3.8, which is a special case of the following statement:…”

Section: Conjectures On (Weakly) Special Degreesmentioning

confidence: 93%

Sets of Special Subvarieties of Bounded Degree

Urbanik¹

2021

Preprint

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Let f : X Ñ S be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base S, and let V " R 2k f˚Zpkq be the integral variation of Hodge structure coming from degree 2k cohomology it induces. Associated to V one has the so-called Hodge locus HLpSq Ă S, which is a countable union of "special" algebraic subvarieties of S parametrizing those fibres of 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 Ă S of degree at most d (with the degree taken relative to a choice of projective compactification S Ă S and very ample line bundle L on 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.

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Effective estimates for the degrees of maximal special subvarieties

Abstract: Let Z be an algebraic subvariety of a Shimura variety. We extend results of the first author to prove an effective upper bound for the degree of a non-facteur maximal special subvariety of Z.

Cited by 3 publications

References 21 publications

Effective computations for weakly optimal subvarieties

Effective computations for weakly optimal subvarieties

Sets of special subvarieties of bounded degree

Sets of Special Subvarieties of Bounded Degree

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