K3 surfaces with a pair of commuting non-symplectic involutions

We study K3 surfaces with a pair of commuting involutions that are non-symplectic with respect to two anti-commuting complex structures that are determined by a hyper-Kähler metric. One motivation for this paper is the role of such $\mathbb{Z}^2_2$-actions for the construction of Riemannian manifolds with holonomy $G_2$. We find a large class of smooth K3 surfaces with such pairs of involutions. After that, we turn our attention to the case that the K3 surface has ADE‑singularities. We introduce a special class of non-symplectic involutions that are suitable for explicit calculations and find 320 examples of pairs of involutions that act on K3 surfaces with a great variety of singularities.

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Received 9 October 2018

Accepted 25 March 2020

Published 17 August 2023