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Complex multiplication and Noether–Lefschetz loci of the twistor space of a K3 surface

Part of: Abelian varieties and schemes Families, fibrations Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  27 July 2023

FRANCESCO VIGANÒ
FRANCESCO VIGANÒ*
Affiliation:
Huxley Building, Imperial College, 180 Queen’s Gate, South Kensington, London SW7 2AZ. Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, 91405 Orsay CEDEX, France. e-mail: f.vigano21@imperial.ac.uk
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Abstract

For an algebraic K3 surface with complex multiplication (CM), algebraic fibres of the associated twistor space away from the equator are again of CM type. In this paper, we show that algebraic fibres corresponding to points at the same altitude of the twistor base ${S^2} \simeq \mathbb{P}_\mathbb{C}^1$ share the same CM endomorphism field. Moreover, we determine all the admissible Picard numbers of the twistor fibres.

MSC classification

Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 14K22: Complex multiplication 14D07: Variation of Hodge structures
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 176 , Issue 1 , January 2024 , pp. 17 - 38
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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