The Twistor Space of a Compact Hypercomplex Manifold is Never Moishezon

Abstract Let $(X,I,J,K)$ be a compact hypercomplex manifold, that is, a smooth manifold $X$ with an action of the quaternion algebra $\langle Id,I,J,K,\rangle ={\mathbb{H}}$ on the tangent bundle $TX$, inducing integrable almost complex structures. For any $(a, b, c) \in S^{2}$, the linear combination $L:= aI + bJ + cK$ defines another complex structure on $X$. This results in a ${\mathbb{C}} P^{1}$-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 $\mathcal{C}$ (in particular, never Kähler and never projective).