Instability of the Fundamental Group for Ricci-Flat Kähler 4-Manifolds

Abstract

We show that the fundamental group is not locally stable under non-collapsed Gromov--Hausdorff convergence, even among closed Ricci-flat Kähler four-manifolds. More precisely, we construct families of Ricci-flat Kähler metrics on $K3$ surfaces and Enriques surfaces that converge to the same compact flat orbifold. Using the same mechanism, we further construct two families of closed locally hyperkähler Ricci-flat four-manifolds, with fundamental groups ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$, respectively, which converge to the same compact flat orbifold.

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