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A decomposition theorem for singular spaces with trivial canonical class of dimension at most five

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Abstract

In this paper we partly extend the Beauville–Bogomolov decomposition theorem to the singular setting. We show that any complex projective variety of dimension at most five with canonical singularities and numerically trivial canonical class admits a finite cover, étale in codimension one, that decomposes as a product of an Abelian variety, and singular analogues of irreducible Calabi–Yau and irreducible holomorphic symplectic varieties.

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Acknowledgements

We would like to thank Jorge V. Pereira and Frédéric Touzet for helpful discussions. The author would also like to thank Cinzia Casagrande for important suggestions. Lemma 4.6 goes back to her. Finally, we would like to thank the referees for their helpful and very detailed reports, and for pointing out several mistakes in a previous version. The author was partially supported by the ALKAGE project (ERC grant Nr. 670846, 2015–2020).

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  1. Institut Fourier, UMR 5582 du CNRS, Université Grenoble Alpes, CS 40700, 38058, Grenoble Cedex 9, France

    Stéphane Druel

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Druel, S. A decomposition theorem for singular spaces with trivial canonical class of dimension at most five. Invent. math. 211, 245–296 (2018). https://doi.org/10.1007/s00222-017-0748-y

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