Albanese maps and fundamental groups of varieties with many rational points over function fields

Javanpeykar, Ariyan; Rousseau, Erwan

doi:10.48550/arxiv.2010.02913

Cited by 3 publications

(4 citation statements)

References 19 publications

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“…We can apply Theorem 1.5 to any pseudo-algebraically hyperbolic projective surface. For example, if X is a projective normal surface of general type which has maximal Albanese dimension, then X is pseudo-algebraically hyperbolic by [Yam15] (see [JR,Theorem 3.9] for details). One can also apply Theorem 1.5 to two-dimensional complete subvarieties of certain locally symmetric varieties.…”

Section: Introductionmentioning

confidence: 99%

Finiteness of non-constant maps over number fields

Javanpeykar¹

2021

Preprint

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Section: Introductionmentioning

confidence: 99%

Finiteness of non-constant maps over number fields

Javanpeykar¹

2021

Preprint

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“…Remark 6.3. In [JR20], it is proved that if there exists a Zariski-dense representation ρ : π 1 (X) → G(C) (G an almost simple algebraic group), then X is not geometrically special. We cannot apply this result here since the monodromy representation ρ : π 1 (X 0 ) → Aut(D) is a priori defined only on X 0 := X \ Supp N where N is the negative part of c 1 (N * F ).…”

Section: Geometric Specialnessmentioning

confidence: 99%

“…On the arithmetic side, we are not able to deal with rational points but rather we study a function field version of Campana's conjecture recently introduced in [JR20]. In this setting, the analogue of potential density is given by geometric specialness as follows.…”

Section: Introductionmentioning

confidence: 99%

Numerically nonspecial varieties

Pereira¹,

Rousseau²,

Touzet³

2021

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Campana introduced the class of special varieties as the varieties admitting no Bogomolov sheaves i.e. rank one coherent subsheaves of maximal Kodaira dimension in some exterior power of the cotangent bundle. Campana raised the question if one can replace the Kodaira dimension by the numerical dimension in this characterization. We answer partially this question showing that a projective manifold admitting a rank one coherent subsheaf of the cotangent bundle with numerical dimension 1 is not special. We also establish the analytic characterization with the non-existence of Zariski dense entire curve and the arithmetic version with non-potential density in the (split) function field setting. Finally, we conclude with a few comments for higher codimensional foliations which may provide some evidence towards a generalization of the aforementioned results. Contents 1. Introduction 1 2. Transversely hyperbolic foliations 3 3. Pull-backs of tautological foliations on irreducible polydisc quotients 10 4. Numerical specialness for rank one subsheaves of Ω 1 12 5. Entire curves 15 6. Geometric specialness 16 7. Higher codimensions 18 References 20

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“…Geometrically special varieties. Recently, Javanpeykar-Rousseau [JR20] defined a new notion of specialness, called geometrically special, which is conjecturally equivalent to Campana's original notion.…”

Section: Intermezzo On Weakly Special Varietiesmentioning

confidence: 99%

A non-Archimedean analogue of Campana's notion of specialness

Morrow¹,

Rosso²

2021

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Let K be an algebraically closed, complete, non-Archimedean valued field of characteristic zero, and let X be a K-analytic space (in the sense of Huber). In this work, we pursue a non-Archimedean characterization of Campana's notion of specialness. We say X is K-analytically special if there exists a connected, finite type algebraic group G/K, a dense open subset U ⊂ G an with codim(G an \ U) 2, and an analytic morphism U → X which is Zariski dense.With this definition, we prove several results which illustrate that this definition correctly captures Campana's notion of specialness in the non-Archimedean setting. These results inspire us to make non-Archimedean counterparts to conjectures of Campana. As preparation for our proofs, we prove auxiliary results concerning the indeterminacy locus of a meromorphic mapping between K-analytic spaces, the notion of pseudo-K-analytically Brody hyperbolic, and extensions of meromorphic maps from smooth, irreducible K-analytic spaces to the analytification of a semi-abelian variety.

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Albanese maps and fundamental groups of varieties with many rational points over function fields

Cited by 3 publications

References 19 publications

Finiteness of non-constant maps over number fields

Finiteness of non-constant maps over number fields

Numerically nonspecial varieties

A non-Archimedean analogue of Campana's notion of specialness

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