Non simply connected HyperKähler 4-manifolds without ALE metrics

Question

In a 1989 paper Peter Kronheimer showed that each simply connected HyperKähler 4-manifold possesses an ALE metric. What do we know about the non-simply connected cases?

Answer 1

I may be missing something :-) I'm not seeing where Kronheimer shows $$ \text{hyper-Kähler + simply connected} \Rightarrow \text{ALE} \tag{?} $$ as you claim. But in [2] he shows that "every hyper-Kähler ALE 4-manifold is isometric to a member of one of the families obtained in [1]." As said members are (I believe) simply connected, this means that $$ \text{hyper-Kähler + ALE} \Rightarrow \text{simply connected} \tag{!} $$ which appears to answer your question. Also of interest: [3] which constructs ALF, ALG, ALH,... metrics on (mostly, but not always, simply connected, hyper-Kähler) 4-manifolds; [4]; and [5], Remark 4.5.

[1] The construction of ALE spaces as hyper-Kähler quotients [2] A Torelli-type theorem for gravitational instantons [3] A Kummer construction for gravitational instantons [4] Quotients of gravitational instantons [5] ALE Ricci-flat Kähler metrics and deformations of quotient surface singularities

Answer 2

"Peter Kronheimer showed that each simply connected HyperKähler 4-manifold possesses an ALE metric."

This is false as stated. K3 surface, or the Taub-NUT space, are simply connected 4-manifolds which are not ALE. What is true is that a complete hyperkaehler 4-manifold with $L^2$-integrable curvature and maximal volume growth is ALE. The "volume growth" condition means that the volume of a ball of radius $r$ grows as $r^4$ (the growth exponent for a Ricci-flat manifold is always bounded by the dimension). For K3 the growth is obviously zero (it is compact); for Taub-NUT the exponent is 3.