I may be missing an obvious example, but here goes...


Let $X$ denote a complex manifold of dimension $n$. If $X$ is Kähler, then the induced metric on any complex submanifold is also Kähler. If instead $X$ is non-Kähler, we can at least say that any coordinate neighborhood $(U,\varphi)$ inherits a Kähler metric from the induced Euclidean metric on an open subset of $\mathbb{C}^n$. If $X$ is already equipped with a metric, note that this metric on $U$ may not be compatible with it. However, I am unsure of what can be said about the metric structure of $U$ if instead we allow something a little more exotic. For example, what if we take $U$ to be the complement of an analytic set? This is the situation in which I am most interested.


My question is:



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Added: Francesco has provided a class of examples in dimensions $\geq 3$ and BS has provided an example in dimension 2 as a comment to Benoît's answer. I am going to write up BS's example, with a few minor additions, as an answer. In case he decides to write his own answer, I'll delete mine and encourage everyone to vote his up.



Kähler metric on a Zariski open subset of a non-Kähler manifold
One can consider the following example.


A Moishezon manifold $M$ is a compact connected complex manifold such that the field of meromorphic functions on $M$ has transcendence degree equal to the complex dimension of $M$. Complex algebraic varieties have this property, but the converse is not true if the dimension is at least $3$.


In 1967, Boris Moishezon showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kaehler metric. Now take a non-projective (hence non-Kaehler) Moishezon manifold $M$. One can prove that such a manifold admits a bimeromorphic modification $\pi \colon \widetilde{M} \to M$ such that $\widetilde{M}$ is projective.


This implies that there exist an analytic subset $S \subset M$ such that $U:=M - S$ is biholomorphic to an open subset of a projective variety (one can take as $S$ a $1$-codimensional analytic subset).


Hence $U$ admits a Kaehler metric.


For further details see the paper by Shanyu Ji [Currents, metrics and Moishezon manifolds](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-158/issue-2/Currents-metrics-and-Moishezon-manifolds/pjm/1102634506.full), Pacific J. Math. Volume **158**, Number 2 (1993), 335-351.


