Non-trivial smooth families of $K3$ surfaces

TL;DR

This paper investigates the topology of the diffeomorphism group of a K3 surface, revealing that its identity component's fundamental group contains an infinitely generated free abelian subgroup, detected via families Seiberg--Witten invariants.

Contribution

It demonstrates the existence of a large free abelian subgroup in the fundamental group of the identity component of the diffeomorphism group of a K3 surface, using families Seiberg--Witten invariants and Einstein metrics.

Findings

The fundamental group of ${ m Diff}_0(X)$ contains a countably infinite rank free abelian subgroup.

Families Seiberg--Witten invariants detect this infinite subgroup.

Moduli space of Einstein metrics plays a crucial role in the proof.

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.