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!
Best Answer
(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 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 for more details.