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Order 3 symplectic automorphisms on K3 surfaces

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Abstract

The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice \(\Lambda _{K3}\), isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps \(\pi _*\) and \(\pi ^*\) induced in cohomology by the rational quotient map \(\pi :X\dashrightarrow Y\), where X is a K3 surface admitting an order 3 symplectic automorphism \(\sigma \) and Y is the minimal resolution of the quotient \(X/\langle \sigma \rangle \); we deduce the relation between the Néron–Severi group of X and the one of Y. Applying these results we describe explicit geometric examples and generalize the Shioda–Inose structures.

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Acknowledgements

We warmly thank Bert van Geemen and Giovanni Mongardi who read a preliminary version of this paper. We are grateful to the referee for the valuable suggestions. The second author is supported by the fellowship INDAM-DP-COFUND-2015 ”INdAM Doctoral Programme in Mathematics and/or Applications Cofunded by Marie Sklodowska-Curie Actions”, Grant Number 713485.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università Statale di Milano, via Saldini 50, 20133, Milano, Italy

    Alice Garbagnati

  2. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 5, 40126, Bologna, Italy

    Yulieth Prieto Montañez

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  1. Alice Garbagnati
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  2. Yulieth Prieto Montañez
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Correspondence to Alice Garbagnati.

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This paper was written during 2020, it cites and and greatly benefits of works by Conway, who died of Covid. We would like to remember him and all the victims of the pandemic. At this time (February 2021) it is estimated that they are 2209195.

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Garbagnati, A., Prieto Montañez, Y. Order 3 symplectic automorphisms on K3 surfaces. Math. Z. 301, 225–253 (2022). https://doi.org/10.1007/s00209-021-02901-9

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