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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

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1435-5345
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Volume 2016, Issue 714

# Some remarks concerning the Grothendieck period conjecture

Jean-Benoît Bost
/ François Charles
• Laboratoire de mathématiques d'Orsay, UMR 8628 du CNRS, Université Paris-Sud, Bâtiment 425, 91405 Orsay cedex, France
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Published Online: 2014-04-30 | DOI: https://doi.org/10.1515/crelle-2014-0025

## Abstract

We discuss various results and questions around the Grothendieck period conjecture, which is a counterpart, concerning the de Rham–Betti realization of algebraic varieties over number fields, of the classical conjectures of Hodge and Tate. These results give new evidence towards the conjectures of Grothendieck and Kontsevich–Zagier concerning transcendence properties of the torsors of periods of varieties over number fields.

Let $\overline{ℚ}$ be the algebraic closure of ℚ in ℂ, let X be a smooth projective variety over $\overline{ℚ}$ and let ${X}_{ℂ}^{\mathrm{an}}$ denote the compact complex analytic manifold that it defines. The Grothendieck period conjecture in codimension k on X, denoted ${\mathrm{GPC}}^{k}\left(X\right)$, asserts that any class α in the algebraic de Rham cohomology group ${H}_{\mathrm{dR}}^{2k}\left(X/\overline{ℚ}\right)$ of X over $\overline{ℚ}$ such that $\frac{1}{{\left(2\pi i\right)}^{k}}{\int }_{\gamma }\alpha \in ℚ$ for every rational homology class γ in ${H}_{2k}\left({X}_{ℂ}^{\mathrm{an}},ℚ\right)$ is the class in algebraic de Rham cohomology of some algebraic cycle of codimension k in X, with rational coefficients.

We notably establish that ${\mathrm{GPC}}^{1}\left(X\right)$ holds when X is a product of curves, of abelian varieties, and of K3 surfaces, and that ${\mathrm{GPC}}^{2}\left(X\right)$ holds for a smooth cubic hypersurface X in ${ℙ}_{\overline{ℚ}}^{5}$. We also discuss the conjectural relationship of Grothendieck classes with the weight filtration on cohomology.

Revised: 2014-02-11

Published Online: 2014-04-30

Published in Print: 2016-05-01

Funding Source: Agence Nationale de la Recherche

Award identifier / Grant number: ANR-2010-BLAN-0119-01

During the preparation of this paper, the first author has partially been supported by the project Positive of the Agence Nationale de la Recherche (grant ANR-2010-BLAN-0119-01) and by the Institut Universitaire de France. Most of this work has been completed while the second author was a member of IRMAR at the University of Rennes 1.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2016, Issue 714, Pages 175–208, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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