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NORMAL FUNCTIONS FOR ALGEBRAICALLY TRIVIAL CYCLES ARE ALGEBRAIC FOR ARITHMETIC REASONS

Part of: Families, fibrations Arithmetic problems. Diophantine geometry Cycles and subschemes

Published online by Cambridge University Press:  16 October 2019

JEFFREY D. ACHTER ,
SEBASTIAN CASALAINA-MARTIN  and
CHARLES VIAL
JEFFREY D. ACHTER
Affiliation:
Colorado State University, Department of Mathematics, Fort Collins, CO 80523, USA; j.achter@colostate.edu
SEBASTIAN CASALAINA-MARTIN
Affiliation:
University of Colorado, Department of Mathematics, Boulder, CO 80309, USA; casa@math.colorado.edu
CHARLES VIAL
Affiliation:
Universität Bielefeld, Fakultät für Mathematik, Germany; vial@math.uni-bielefeld.de

Abstract

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For families of smooth complex projective varieties, we show that normal functions arising from algebraically trivial cycle classes are algebraic and defined over the field of definition of the family. In particular, the zero loci of those functions are algebraic and defined over such a field of definition. This proves a conjecture of Charles.

MSC classification

Primary: 14C25: Algebraic cycles 14C30: Transcendental methods, Hodge theory , Hodge conjecture 14D07: Variation of Hodge structures 14D10: Arithmetic ground fields (finite, local, global) 14G99: None of the above, but in this section
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e36
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) 2019

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