Let $X$ be a compact complex non-Kähler manifold, then what conditions do we need to make it has a Kähler deformation? that is to say it can be deformed to a Kähler manifold.
Obviously not all the compact complex manifolds can be deformed to Kähler ones, for example, the Hopf surface, but certainly, there exist some non-Kähler manifolds which can be deformed to Kähler ones. For example, Hironaka has provided an example that except the central fiber, all the other fibers are projective manifolds, and the central fiber is a non-Kähler Moishezon manifold, so, conversely, for this Moishezon manifold, we can say it has a Kähler (even projective) deformation.
Then, are there any other examples of non-Kähler manifolds which has a Kähler deformation? or even a projective deformation? For example does a $partialbarpartial$-manifolds with trivial canonical bundle has a Kähler deformation? Has anyone think about it before? And what’s the latest progress on it?