Twistor spaces and compact manifolds admitting both K\"ahler and non-K\"ahler structures

TL;DR

This paper reviews twistor techniques and discusses the existence of compact manifolds that admit both Kähler and non-Kähler complex structures, highlighting the role of twistor spaces in such constructions.

Contribution

It demonstrates that the quaternion twistor space of a hyperkähler manifold can admit both Kähler and non-Kähler structures, extending previous examples.

Findings

Quaternion twistor space of hyperkähler manifolds admits both structures

Revisits classical examples of manifolds with mixed complex structures

Connects twistor theory with complex geometry

Abstract

In this expository paper we review some twistor techniques and recall the problem of finding compact differentiable manifolds that can carry both K\"ahler and non-K\"ahler complex structures. Such examples were constructed independently by M. Atiyah, A. Blanchard and E. Calabi in the $1950$'s. In the $1980$'s V. Tsanov gave an example of a simply connected manifold that admits both K\"ahler and non-K\"ahler complex structures - the twistor space of a $K3$ surface. Here we show that the quaternion twistor space of a hyperk\"ahler manifold has the same property.