The Nielsen realization problem for K3 surfaces

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 S₄ action by isometries of some Ricci-flat metric on M that preserves no complex structure. We also show that the subgroups of Diff(M) of a given prime order p which fix pointwise some positive-definite 3-plane in H₂(M; R) and preserve some complex structure on M form a single conjugacy class in Diff(M) (it is known that then p ∈ {2, 3, 5, 7}).