The Nielsen realization problem for K3 surfaces

Abstract

The smooth (resp. metric and complex) Nielsen Realization Problem for K3 surfaces $M$ asks: when can a finite group $G$ of mapping classes of $M$ be realized by a finite group of diffeomorphisms (resp. isometries of a Ricci-flat metric, or automorphisms of a complex structure)? We solve the metric and complex versions of Nielsen Realization, and we solve the smooth version for involutions. Unlike the case of 2-manifolds, some $G$ are realizable and some are not, and the answer depends on the category of structure preserved. In particular, Dehn twists are not realizable by finite order diffeomorphisms. We introduce a computable invariant $L_G$ that determines in many cases whether $G$ is realizable or not, and apply this invariant to construct an $S_4$ action by isometries of some Ricci-flat metric on $M$ that preserves no complex structure.

We also show that the subgroups of $\mathrm{Diff}(M)$ of a given prime order $p$ which fix pointwise some positive-definite 3-plane in $H^2(M; \mathbb{R})$ and preserve some complex structure on $M$ form a single conjugacy class in $\mathrm{Diff}(M)$ (it is known that then $p \in \{2, 3, 5, 7\}$).