Birational aspects in the study of LCK manifolds

We study the behaviour of locally conformally Kahler (LCK for short) manifolds under birational transformations. We show that the blow-up of an LCK manifold $X$ along a subvariety $Y$ is LCK iff $Y$ is globally conformallly Kahler (GCK). Using the same methods, we also show that a twistor space is LCK iff it is GCK. We will also adress a number of open questions.
This is joint work with L. Ornea and M. Verbitsky.