Could you give me an example of a compact Kähler manifold which analytically deforms to a non Kähler one?


For example, there is no hope to find a complex structure on a Hopf manifold in order to make it Kähler because of topological obstructions (the second Betti number is zero).


For instance, I think that the Iwasawa manifold should not have topological obstructions.


Of course, the algebraic counterpart of my question has an affirmative answer: complex tori of dimension greater than one give examples of manifolds that can be analytically deformed from an algebraic one to a non algebraic one (but still Kähler). 


Thanks in advance!



Deform a compact Kähler manifold to a non Kähler one
(Just so this question has an answer. All manifolds are compact.)


In **dimension one** every deformation of Kähler manifolds is Kähler because every Riemann surface is Kähler. 


In **dimension two** the same is true but for less trivial reasons. A two-dimensional complex manifold is Kähler if and only if its first Betti number is even, which is purely a topological condition. The result then follows from the fact that the fibres of a deformation are diffeomorphic. 


In **dimension three**, the result is no longer true; that is, there is a deformation of Kähler manifolds such that the central fibre is not Kähler. As Gunnar Magnusson pointed out in the comments, Hironaka gave an example of such a deformation in his [paper](http://www.jstor.org/stable/1970426) *An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian Complex Structures*. In fact, the construction used gives rise to a whole host of interesting phenomena in dimension three including the existence of a Moishezon manifold which is not projective. See the associated page on [Wikipedia](http://en.wikipedia.org/wiki/Hironaka%27s_example) for more details.


