Abstract Let (X, I, J, K) be a compact hypercomplex manifold, that is, a smooth manifold X with an action of the quaternion algebra Id, I, J, K, =H on the tangent bundle TX, inducing integrable almost complex structures. For any (a, b, c) S^2, the linear combination L: = aI + bJ + cK defines another complex structure on X. This results in a C P^1-family of complex structures called the twistor family. Its total space is called the twistor space. We show that the twistor space of a compact hypercomplex manifold is never Moishezon and, moreover, it is never Fujiki class C (in particular, never Kähler and never projective).
Yulia Gorginyan (Fri,) studied this question.