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On the integral Hodge conjecture for real varieties, I

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Abstract

We formulate the “real integral Hodge conjecture”, a version of the integral Hodge conjecture for real varieties, and raise the question of its validity for cycles of dimension 1 on uniruled and Calabi–Yau threefolds and on rationally connected varieties. We relate it to the problem of determining the image of the Borel–Haefliger cycle class map for 1-cycles, with the problem of deciding whether a real variety with no real point contains a curve of even geometric genus and with the problem of computing the torsion of the Chow group of 1-cycles of real threefolds. New results about these problems are obtained along the way.

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Notes

  1. We mean Calabi–Yau in the sense that \(K_X\simeq {{\mathscr {O}}}_X\) and \(H^1(X,{{\mathscr {O}}}_X)=H^2(X,{{\mathscr {O}}}_X)=0\). Totaro has announced a proof of the integral Hodge conjecture for complex projective threefolds X such that \(K_X \simeq {{\mathscr {O}}}_X\). Abelian threefolds are dealt with in [58, Corollary 3.1.9]. The hypothesis that \(K_X \simeq {{\mathscr {O}}}_X\) cannot be weakened to \(K_X\) being torsion in view of [21, Theorem 0.1].

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Acknowledgements

Krasnov and van Hamel were the first to approach algebraic cycles on real varieties in a systematic way through the study of the cycle class map into the equivariant integral singular cohomology of the complex locus. We would like to emphasise the importance of their work to the development of the subject considered in this article. In addition, we thank the referee for their careful work and for many suggestions which helped improve the exposition.

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  1. Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000, Strasbourg, France

    Olivier Benoist

  2. Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, 75230, Paris Cedex 05, France

    Olivier Wittenberg

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Benoist, O., Wittenberg, O. On the integral Hodge conjecture for real varieties, I. Invent. math. 222, 1–77 (2020). https://doi.org/10.1007/s00222-020-00965-8

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