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Non-symplectic automorphisms of odd prime order on manifolds of \(K3^{\left[ n\right] }\)-type

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We contribute to the classification of non-symplectic automorphisms of odd prime order on irreducible holomorphic symplectic manifolds which are deformations of Hilbert schemes of any number n of points on K3 surfaces, extending results already known for \(n=2\). In particular, we study the properties of the invariant lattice of the automorphism (and its orthogonal complement) inside the second cohomology lattice of the manifold. We also explain how to construct automorphisms with fixed action on cohomology: in the cases \(n=3,4\) the examples provided realize all admissible actions in our classification. For \(n=4\), we present a construction of non-symplectic automorphisms on the Lehn–Lehn–Sorger–van Straten eightfold, which come from automorphisms of the underlying cubic fourfold.

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Acknowledgements

Alberto Cattaneo is grateful to Max Planck Institute for Mathematics in Bonn for its hospitality and financial support. The authors thank Samuel Boissière, Simon Brandhorst, Andrea Cattaneo, Alice Garbagnati, Robert Laterveer and Giovanni Mongardi for many helpful discussions, as well as Christian Lehn, Manfred Lehn and Gregory Sankaran for their useful remarks and explanations. The authors are also extremely grateful to Alessandra Sarti and Bert van Geemen for their precious suggestions, and to the anonymous referee for many valuable comments on how to improve the paper.

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  1. Alberto Cattaneo

    Present address: Mathematisches Institut and Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

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  1. Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133, Milan, Italy

    Chiara Camere & Alberto Cattaneo

  2. Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

    Alberto Cattaneo

  3. Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Université de Poitiers, Bât. H3 - Site du Futuroscope TSA 61125, 11 bd Marie et Pierre Curie, 86073, POITIERS Cedex 9, France

    Alberto Cattaneo

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Appendix A: Invariant and co-invariant lattices for \(n=3,4\), \(p=3\)

Appendix A: Invariant and co-invariant lattices for \(n=3,4\), \(p=3\)

The two tables in this appendix list all admissible triples (p, m, a) (see Definition 3.10) and the corresponding isometry classes for the co-invariant lattice \(S \subset H^2(X, \mathbb {Z})\) and the invariant lattice \(T \subset H^2(X, \mathbb {Z})\) of non-symplectic automorphisms of order \(p = 3\) on manifolds X of \(K3^{[n]}\)-type, for \(n = 3, 4\). This classification is discussed in Sect. 3.4.

The symbol \(\clubsuit \) denotes the cases which can be realized by natural automorphisms (see Sect. 4.1). The cases marked with \(\natural \) (respectively, \(\diamondsuit \)) correspond to admissible triples that admit a realization by induced automorphisms on moduli spaces of ordinary (respectively, twisted) sheaves on K3 surfaces, but not by natural automorphisms (see Sect. 5). Finally, the admissible triple (3, 11, 0) for \(n=4\) (marked with the symbol \(\bigstar \)) is realized by the automorphism constructed in Sect. 6.1 on a ten-dimensional family of Lehn–Lehn–Sorger–van Straten eightfolds (see also Sect. 4.3).

We recall that in the following tables, as in the rest of the paper, the root lattices \(A_2, E_6, E_8\) are defined as negative definite.

Table 1 \(n=3\), \(p=3\). See Sect. 3.4 for the definition of the lattice \(\Omega \)
Full size table
Table 2 \(n = 4\), \(p = 3\)
Full size table

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Camere, C., Cattaneo, A. Non-symplectic automorphisms of odd prime order on manifolds of \(K3^{\left[ n\right] }\)-type. manuscripta math. 163, 299–342 (2020). https://doi.org/10.1007/s00229-019-01163-4

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