The twistor space of a compact hypercomplex manifold is never Moishezon

Abstract

Let (X,I,J,K) be a compact hypercomplex manifold, i.e. a smooth manifold X with an action of the quaternion algebra (Id,I,J,K) 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 $C{P}^1$-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 Kahler and never projective).

16 pages, version 1.0