TITLE: Diffeomorphic Kähler manifolds with different Hodge numbers
QUESTION [29 upvotes]: This question made me wonder about the following:
Are there orientedly diffeomorphic Kähler manifolds with different Hodge numbers?
It seems that this would require that those manifolds are not deformation equivalent.
However, there are examples by Catanese and Manetti that that happens already for smooth projective surfaces.

REPLY [13 votes]: A few years after this post appeared, the question of oriented diffeomorphism invariance of Hodge and Chern numbers of smooth, projective varieties was settled completely by Kotschick and Schreieder in this paper. In fact, they even refer to this post(!).