The Twistor Space of a Compact Hypercomplex Manifold is Never Moishezon

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) ∈ S², the linear combination $L:= aI + bJ + cK$ defines another complex structure on X. This results in a C P¹-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).