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Attractors are not algebraic

Part of: Surfaces and higher-dimensional varieties Arithmetic algebraic geometry Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  18 April 2024

Yeuk Hay Joshua Lam Open the ORCID record for Yeuk Hay Joshua Lam [Opens in a new window]  and
Arnav Tripathy
Yeuk Hay Joshua Lam
Affiliation:
Institut für Mathematik–Alg.Geo., Humboldt Universität Berlin, Rudower Chaussee 25, Berlin, Germany joshua.lam@hu-berlin.de
Arnav Tripathy
Affiliation:
Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Huairou District, Beijing 101408, PR China arnav.tripathy@gmail.com
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Abstract

The attractor conjecture for Calabi–Yau moduli spaces predicts the algebraicity of the moduli values of certain isolated points picked out by Hodge-theoretic conditions. Using tools from transcendence theory, we provide a family of counterexamples to the attractor conjecture in almost all odd dimensions conditional on a specific case of the Zilber–Pink conjecture in unlikely intersection theory; these Calabi–Yau manifolds were first studied by Dolgachev. We also give constructions of new families of Calabi–Yau varieties, analogous to the mirror quintic family, with all middle Hodge numbers equal to one, which would also give counterexamples to the attractor conjecture.

Keywords

Calabi–Yau varietiesZilber–Pink conjecturetranscendence

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties
Secondary: 14J32: Calabi-Yau manifolds 11J81: Transcendence (general theory)
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 5 , May 2024 , pp. 1073 - 1100
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Copyright
© 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

AT was supported under NSF MSPRF grant 1705008 during the course of this work.

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