Advanced Studies in Pure Mathematics

On derived categories of K3 surfaces, symplectic automorphisms and the Conway group

Daniel Huybrechts

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In this note we interpret a recent result of Gaberdiel et al [7] in terms of derived equivalences of K3 surfaces. We prove that there is a natural bijection between subgroups of the Conway group $C\!o_0$ with invariant lattice of rank at least four and groups of symplectic derived equivalences of $\mathrm{D}^{\mathrm{b}}(X)$ of projective K3 surfaces fixing a stability condition.

As an application we prove that every such subgroup $G\subset C\!o_0$ satisfying an additional condition can be realized as a group of symplectic automorphisms of an irreducible symplectic variety deformation equivalent to $\mathrm{Hilb}^n(X)$ of some K3 surface.

Article information

Development of Moduli Theory — Kyoto 2013, O. Fujino, S. Kondō, A. Moriwaki, M. Saito and K. Yoshioka, eds. (Tokyo: Mathematical Society of Japan, 2016), 387-405

Received: 25 September 2013
Revised: 28 July 2014
First available in Project Euclid: 4 October 2018

Permanent link to this document euclid.aspm/1538622437

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J28: $K3$ surfaces and Enriques surfaces 14C30: Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture


Huybrechts, Daniel. On derived categories of K3 surfaces, symplectic automorphisms and the Conway group. Development of Moduli Theory — Kyoto 2013, 387--405, Mathematical Society of Japan, Tokyo, Japan, 2016. doi:10.2969/aspm/06910387.

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