The Diffeomorphism Group of a K3 Surface and Nielsen Realization

The Nielsen Realization problem asks when the group homomorphism Diff(M) → π0Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enough then no section exists over the entire mapping class group. We prove the first nonexistence theorem of this type in dimension 4: if M is a smooth closed oriented 4-manifold which contains a K3 surface as a connected summand then no section exists over the whole of the mapping class group. This is done by showing that certain obstructions lying in the rational cohomology of Bπ0Diff(M) are nonzero. We detect these classes by showing that they are nonzero when pulled back to the moduli space of Einstein metrics on a K3 surface.