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

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.