Non-trivial smooth families of K3 surfaces

Let X be a complex K3 surface, Diff(X) the group of diffeomorphisms of X and Diff(0)(X) the identity component. We prove that the fundamental group of Diff(0)(X) contains a free abelian group of countably infinite rank as a direct summand. The summand is detected using families Seiberg-Witten invariants. The moduli space of Einstein metrics on X is used as a key ingredient in the proof.