  The Nielsen Realization problem asks when the group homomorphism from Diff(M)
to pi_0 Diff(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 B(pi_0 Diff(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.
