Non-trivial smooth families of $K3$ surfaces

Abstract

Let $X$ be a complex $K3$ surface, ${\rm Diff}(X)$ the group of diffeomorphisms of $X$ and ${\rm Diff}_0(X)$ the identity component. We prove that the fundamental group of ${\rm Diff}_0(X)$ 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.