doi:10.1093/imrn/rns128 Blow-ups of Locally Conformally Kähler Manifolds

A locally conformally Khler (LCK) manifold is a complex manifold which admits a cover-ing endowed with a Kähler metric with respect to which the covering group acts through homotheties. We show that the blow-up of a compact LCKmanifold along a complex sub-manifold admits an LCK structure if and only if this submanifold is globally conformally Kähler. We also prove that a twistor space (of a compact four-manifold, a quaternion-Kähler manifold, or a Riemannian manifold) cannot admit an LCK metric, unless it is Kähler.