Compact complex threefolds which are Kähler outside a smooth rational curve

Let M be a compact complex manifold containing an irreducible curve C such that M — C is Kähler; in this paper we study the link between some cohomological properties of C and the obstructions to the existence of a Kähler metric on the whole of M. In particular we get that, if M is not Kähler, then C is a $ \left({\partial + \bar \partial} \right) $ –exact current, or there exists a positive current S of bidimension (1, 1) such that $ \partial \bar \partial S = 0,\,\chi _C S = 0 $ and S + C is $ \left({\partial + \bar \partial} \right) $ –exact. If C is a smooth rational curve, more precise results are given in connection with the normal bundle N_C|M.