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A well-known example in the deformation theory of compact complex manifolds is the one given by Hironaka in his 1962 paper An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian Complex Structures. The construction gives rise to many interesting phenomena (see the associated Wikipedia article), but the one that interests me most is that this was the first example of a deformation of Kähler manifolds which has limiting fibre which is not Kähler.

Aside from Hironaka's example, are there any other explicit deformations of compact Kähler manifolds such that the central fibre is not Kähler?

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  • $\begingroup$ When fibers are projective, on $\pi: X\to\Delta$ then the central fiber $X_0$ may not be Kahler , it is Moishezon manifold due to Dan Popovici link.springer.com/article/10.1007/s00222-013-0449-0 , Daniel Barlet says that this result is due to Hironaka . But Barlet gave a lot of interesting examples here arxiv.org/pdf/1501.01129.pdf $\endgroup$
    – user21574
    Commented Dec 1, 2017 at 4:24

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