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Nonspecial varieties and generalised Lang–Vojta conjectures

Part of: Diophantine approximation, transcendental number theory Holomorphic functions of several complex variables Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  10 February 2021

Erwan Rousseau ,
Amos Turchet Open the ORCID record for Amos Turchet [Opens in a new window]  and
Julie Tzu-Yueh Wang
Erwan Rousseau
Affiliation:
Institut Universitaire de France & Aix-Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France; E-mail: erwan.rousseau@univ-amu.fr Freiburg Institute for Advanced Studies, University of Freiburg, Albertstr. 19, 79104Freiburg, Germany
Amos Turchet
Affiliation:
Dipartimento di Matematica e Fisica, Universitá degli studi Roma 3, L.go S. L. Murialdo 1, 00146Roma, Italy; E-mail: amos.turchet@uniroma3.it
Julie Tzu-Yueh Wang
Affiliation:
Institute of Mathematics, Academia Sinica No. 1, Sec. 4, Roosevelt Road Taipei10617, Taiwan; E-mail: jwang@math.sinica.edu.tw

Abstract

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We construct a family of fibred threefolds $X_m \to (S , \Delta )$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,\Delta )$ of general type. Following Campana, the threefolds $X_m$ are called weakly special but not special. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if m is big enough, the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conjecture. We prove our results by adapting the recent method of Ru and Vojta. We also formulate some generalisations of known conjectures on exceptional loci that fit into Campana’s program and prove some cases over function fields.

Keywords

Campana’s conjecturesFunction fieldsNevanlinna TheoryOrbifoldsHyperbolicity

MSC classification

Primary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights
Secondary: 11J97: Analogues of methods in Nevanlinna theory (work of Vojta et al.) 14G05: Rational points 32A22: Nevanlinna theory (local); growth estimates; other inequalities
Type
Algebraic and Complex Geometry
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e11
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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