## Advanced Studies in Pure Mathematics

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

Daniel Huybrechts

#### Abstract

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

Dates
Revised: 28 July 2014
First available in Project Euclid: 4 October 2018

https://projecteuclid.org/ euclid.aspm/1538622437

Digital Object Identifier
doi:10.2969/aspm/06910387

Mathematical Reviews number (MathSciNet)
MR3586514

Zentralblatt MATH identifier
1386.14137

#### Citation

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. https://projecteuclid.org/euclid.aspm/1538622437