Blow-ups of Locally Conformally Kähler Manifolds

Abstract

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