The Nielsen realization problem for K3 surfaces (Q6562496)

A K3-surface is a closed, simply connected complex surface that admits a nowhere vanishing holomorphic 2-form; all K3-surfaces are diffeomorphic 4-manifolds (``the K3-manifold''). Since K3-surfaces admit both complex structures and Ricci-flat metrics, there are basically three versions of the Nielsen realization problem, each stronger than the previous one: realization of finite subgroups of the mapping class group \(\pi_0(\mathrm{Diff}(M))\) of a K3-surface \(M\) by diffeomorphisms (\textit{smooth version}), by isometries of some Ricci-flat metric on \(M\) (\textit{metric version}), and finally by complex automorphisms (\textit{complex version}). In the present paper, the authors solve the metric and complex Nielsen realization problems for finite subgroups \(G\) of mapping classes, and the smooth version for involutions; in particular, Dehn twists are not realizable by finite order diffeomorphisms. ``We introduce a computable invariant \(\mathbf{L}_G\) 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 \(p \in \{2,3,5,7\}\)).''