In this expository paper we review some twistor techniques and recall the problem of finding compact differentiable manifolds that can carry both Kähler and non-Kähler complex structures. Such examples were constructed independently by Atiyah, Blanchard and Calabi in the 1950’s. In the 1980’s Tsanov gave an example of a simply connected manifold that admits both Kähler and non-Kähler complex structures - the twistor space of a $K3$ surface. Here we show that the quaternion twistor space of a hyperkähler manifold has the same property.
"Twistor Spaces and Compact Manifolds Admitting Both Kähler and Non-Kähler Structures." J. Geom. Symmetry Phys. 46 25 - 35, 2017. https://doi.org/10.7546/jgsp-46-2017-25-35