On the integral Hodge conjecture for real abelian threefolds

Olivier de Gaay Fortman

doi:10.1515/crelle-2023-0082

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On the integral Hodge conjecture for real abelian threefolds

Veröffentlicht/Copyright: 12. Januar 2024

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2024 Heft 807

Abstract

We prove the real integral Hodge conjecture for several classes of real abelian threefolds. For instance, we prove the property for real abelian threefolds whose real locus is connected, and for real abelian threefolds A which are the product A = B × E of an abelian surface B and an elliptic curve E with connected real locus E ⁢ ( ℝ ) . Moreover, we show that every real abelian threefold satisfies the real integral Hodge conjecture modulo torsion, and reduce the principally polarized case to the Jacobian case.

Funding statement: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754362.

Acknowledgements

This paper is based on results that appear in the last chapter of my PhD thesis [12]. I thank my advisor Olivier Benoist for the great guidance he has given me over the past three years. In particular, I owe him much for drawing my attention to the real integral Hodge conjecture, and for the many discussions we had concerning this project. I thank Nicolas Tholozan for several discussions on moduli of real abelian varieties, and in particular for explaining to me how to prove Theorem 1.5. Moreover, I thank Fabrizio Catanese, François Charles and Javier Fresán for stimulating conversations, and I thank Olivier Benoist and Olivier Wittenberg for useful comments on an earlier version of this paper.

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Received: 2022-12-05

Revised: 2023-10-15

Published Online: 2024-01-12

Published in Print: 2024-02-01

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