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Motives of derived equivalent K3 surfaces

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Abstract

We observe that derived equivalent K3 surfaces have isomorphic Chow motives. The result holds more generally for arbitrary surfaces, as pointed out by Charles Vial.

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Notes

  1. For K3 surfaces the isomorphism is indeed between integral Chow groups, but this is not true for other types of varieties.

  2. or, weaker, in \(K_0(\mathrm{Var}(k))/(\ell )\) which in characteristic zero is isomorphic to the Grothendieck ring of stable birational classes of smooth projective varieties \({\mathbb {Z}}[\mathrm{SB}(k)]\), see [18].

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Acknowledgements

I am very grateful to Charles Vial for answering my questions and to him and Andrey Soldatenkov for comments on a first version of this note and helpful suggestions. Thanks also to Jeff Achter who insisted that the result should hold without any restriction on the field and to the referee for a thorough reading and constructive suggestions.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    D. Huybrechts

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  1. D. Huybrechts
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Correspondence to D. Huybrechts.

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Communicated by Daniel Greb.

The author is supported by the SFB/TR 45 ‘Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’ of the DFG (German Research Foundation).

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Huybrechts, D. Motives of derived equivalent K3 surfaces. Abh. Math. Semin. Univ. Hambg. 88, 201–207 (2018). https://doi.org/10.1007/s12188-017-0182-5

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