Non-symplectic automorphisms of odd prime order on manifolds of $$K3^{\left[ n\right] }$$ K 3 n -type
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Chiara Camere
© Springer-Verlag GmbH Germany, part of Springer Nature 2019
· Alberto Cattaneo
Non-symplectic automorphisms of odd prime order on manifolds of K 3[n] -type Received: 3 April 2018 / Accepted: 8 November 2019 Abstract. 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 K 3 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.
1. Introduction The study of automorphisms of K 3 surfaces has been a very active research field for decades. The global Torelli theorem allows to reconstruct automorphisms of a K 3 surface from Hodge isometries of H 2 (, Z) preserving the intersection product; this link, together with the seminal works of Nikulin [55,56], provided the instruments to investigate finite groups of automorphisms on K 3’s. In recent years, the interest in automorphisms has extended from K 3 surfaces to manifolds which generalize them in higher dimension, namely irreducible holomorphic symplectic (IHS) varieties. Results by Huybrechts [36], Markman [47] and Verbitsky [65], which provide an analogous of the Torelli theorem for these manifolds, allow us to use similar methods, studying the action of an automorphism on the second cohomology group with integer coefficients (which carries again a lattice structure, C. Camere (B) · A. Cattaneo: Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milan, Italy e-mail: [email protected] A. Cattaneo: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany e-mail: [email protected] A. Cattaneo: 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 A. Cattaneo: Present address: Mathematisches Institut and Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany Mathematics Subject Classification: 14J50 · 14C05 · 14C34
https://doi.org/10.1007/s00229-019-01163-4
C. Camere, A. Cattaneo
provided by the Beauville–Bogomolov–Fujiki quadratic form; see [34, Part III] for further references). A great number of results are known for automorphisms of prime order on IHS fourfolds that are deformations of the Hilbert scheme of two points on a K 3 surface (so-called manifolds of K 3[2] -type). The symplectic case (i.e. automorphisms which preserve the sym
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