The twistor space of a compact hypercomplex manifold is never Moishezon

TL;DR

The paper proves that the twistor space associated with any compact hypercomplex manifold cannot be Moishezon, Fujiki class C, Kahler, or projective, highlighting fundamental geometric restrictions of these structures.

Contribution

It establishes that the twistor space of a compact hypercomplex manifold is never Moishezon, extending to it being never Fujiki class C, Kahler, or projective, which was previously unknown.

Findings

Twistor space of a compact hypercomplex manifold is never Moishezon.

Twistor space is never Fujiki class C.

Twistor space is never Kahler or projective.

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).