Bimeromorphic geometry of LCK manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold $M$ which has a Kähler structure on its cover, such that the deck transform group acts on it by homotheties. Assume that the Kähler form is exact on the minimal Kähler cover of $M$. We prove that any bimeromorphic map $M^{\mathrm{\prime }}\to M$ is in fact holomorphic; in other words, $M$ has a unique minimal model. This can be applied to a wide class of LCK manifolds, such as the Hopf manifolds, their complex submanifolds and to OT manifolds.