Non-trivial smooth families of $K3$ surfaces

Abstract

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

22 pages, accepted version. To appear in Math. Ann

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