# Metric connection on $\mathbb{R}^4$ that is locally Kähler but not globally Kähler

in a comment to this question When can a Connection Induce a Riemannian Metric for which it is the Levi-Civita Connection?

Robert Bryant mentions that it is possible to construct a metric connection on $\mathbb{R}^4$ that is locally Kähler, but not Kähler. I haven't been able to figure out how to construct such connection, can anyone give me a hint?

Thanks.

• Hint: inside $SO(4)$ there are two copies of $U(2)$ whose product is $SO(4)$ see e.g Van Elfrinkhof's formula in en.wikipedia.org/wiki/…. – Holonomia Jan 20 '18 at 11:44