Torsion Points of Sections of Lagrangian Torus Fibrations and the Chow Ring of Hyper-Kähler Manifolds

Voisin, Claire

doi:10.1007/978-3-319-94881-2_10

Cited by 16 publications

(23 citation statements)

References 34 publications

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“…Betti map was first studied and used in [Zan12]. Then it was used to study the relative Manin-Mumford conjecture by Bertrand, Corvaja, Masser, Pillay, and Zannier in a series of works [MZ12,MZ14,MZ15,BMPZ16,CMZ18,MZ18], to prove the geometric Bogomolov conjecture over char 0 by Gao and Habegger [GH19] and Cantat, Gao, Habegger and Xie [CGHX20] with (1.1) for l = dim X − dim S, and to prove the denseness of torsion points on sections of Lagrangian fibrations by Voisin [Voi18] using André, Corvaja and Zannier's result [ACZ20] on (1.1) for l = g ≤ 4.…”

Section: Z Gaomentioning

confidence: 99%

“…Part (i) is Theorem 9.2; see Remark 9.3 for (i)(b). Part (i)(a) for l = g, combined with [ACZ20, Proposition 2.1.1], shows that [Voi18, Theorem 0.3] holds without the dimension assumption because by Lemma 4.5 of [Voi18] the abelian scheme in question is geometrically simple. Part (ii) is constructed in Example 9.4; it is closely related to [ACZ20, Remark 6.2.1].…”

Section: Acz Question Assume That A/s Has No Fixed Part Over Any Finmentioning

confidence: 99%

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Generic rank of Betti map and unlikely intersections

Gao¹

2020

Compositio Math.

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Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$ . For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$ , the real torus of dimension $2g$ , by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.

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Section: Z Gaomentioning

confidence: 99%

Section: Acz Question Assume That A/s Has No Fixed Part Over Any Finmentioning

confidence: 99%

Generic rank of Betti map and unlikely intersections

Gao¹

2020

Compositio Math.

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“…(vi) Still from a different perspective, C. Voisin [36] considered very recently a closely related problem, this time motivated by the investigation of Chow groups. In the context of Lagrangian fibrations on hyperkähler manifolds, she uses methods of her own for the case g ≤ 2, and our Corollary 2.2.4 to prove the desired conclusion for g ≤ 4.…”

Section: 3mentioning

confidence: 99%

The Betti map associated to a section of an abelian scheme

2020

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Given a point ξ on a complex abelian variety A, its abelian logarithm can be expressed as a linear combination of the periods of A with real coefficients, the Betti coordinates of ξ. When (A, ξ) varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often ξ takes a torsion value (for instance, Manin's theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when ξ is a torsion section).We compute this rank in terms of the rank of a certain contracted form of the Kodaira-Spencer map associated to (A, ξ) (assuming A without fixed part, and Zξ Zariski-dense in A), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension ≤ 3, and study in detail the case of jacobians of families of hyperelliptic curves.Our main application, obtained in collaboration with Z. Gao, states that if A → S is a principally polarized abelian scheme of relative dimension g which has no non-trivial endomorphism (on any finite covering), and if the image of S in the moduli space Ag has dimension at least g, then the Betti map of any non-torsion section ξ is generically a submersion, so that ξ −1 Ators is dense in S(C).

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“…This seems an interesting matter in its own, which recently attracted some interest from differenty authors and opens several questions that we shall discuss in future. In a recent work by C. Voisin [23], the author implicitely used the Betti map in order to prove the density of torsion points in certain abelian schemes, arising from Lagrangian fibrations.…”

Section: Our First Theorem Ismentioning

confidence: 99%

“…Further warm thanks go to D. Bertrand for his interest and comments which helped our presentation. We are also grateful to C. Voisin for sending us her recent preprint [23] and for a helpful correspondence with the third author.…”

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confidence: 99%

Torsion hypersurfaces on abelian schemes and Betti coordinates

2017

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In this paper we extend to arbitrary complex coefficients certain finiteness results on Unlikely intersections linked to torsion in abelian surface schemes over a curve, which have been recently proved for the case of algebraic coefficients; in this way we complete the solution of Zilber-Pink conjecture for abelian surface schemes over a curve. As experience has shown also in previous cases, the extension from algebraic to complex coefficients often requires entirely new arguments, whereas simple specialization arguments fail. The outcome gives as a byproduct new finiteness results when the base of the scheme has arbitrary dimension; another consequence is a proof of an expectation of Mazur concerning the structure of the locus in the base when a given section is torsion. Finally, we show the link with an old work of Griffiths and Harris on a higher dimensional extension of a theorem of Poncelet.

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Torsion Points of Sections of Lagrangian Torus Fibrations and the Chow Ring of Hyper-Kähler Manifolds

Cited by 16 publications

References 34 publications

Generic rank of Betti map and unlikely intersections

Generic rank of Betti map and unlikely intersections

The Betti map associated to a section of an abelian scheme

Torsion hypersurfaces on abelian schemes and Betti coordinates

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