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A pencil of Enriques surfaces with non-algebraic integral Hodge classes

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Abstract

We prove that there exists a pencil of Enriques surfaces defined over \({\mathbb {Q}}\) with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question of Murre on the universality of the Abel-Jacobi maps in codimension three.

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Notes

  1. We use Grothendieck’s notation for projective bundles: for a vector bundle \({\mathcal {E}}\), \({\mathbb {P}}({\mathcal {E}})\) paramterizes one-dimensional quotients of \({\mathcal {E}}\).

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Acknowledgements

The authors would like to thank Olivier Benoist, Jørgen Vold Rennemo, Jason Starr, and Claire Voisin for interesting discussions. The second author wishes to thank his advisor Lawrence Ein for constant support and warm encouragement. JCO was supported by the Research Council of Norway project no. 250104. FS was supported by the NSF Grant No. DMS-1801870. This project started while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019 semester.

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Authors and Affiliations

  1. Department of Mathematics, University of Oslo, Box 1053, Blindern, 0316, Oslo, Norway

    John Christian Ottem

  2. Department of Mathematics, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL, USA

    Fumiaki Suzuki

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  1. John Christian Ottem
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  2. Fumiaki Suzuki
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Correspondence to Fumiaki Suzuki.

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Communicated by Vasudevan Srinivas.

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Ottem, J.C., Suzuki, F. A pencil of Enriques surfaces with non-algebraic integral Hodge classes. Math. Ann. 377, 183–197 (2020). https://doi.org/10.1007/s00208-020-01969-8

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