Compact complex threefolds which are Kahler outside a smooth rational curve

Let M be a compact complex manifold containing an irreducible curve C such that M - C is Kahler; in this paper we study the link between some cohomolo gical properties of C and the obstructions to the existence of a Kahler met ric on the whole of M. In particular we get that, if M is not Kahler, then C is a (partial derivative + <(partial derivative)over bar>) - exact curren t, or there exists a positive current S of bidimension (1, 1) such that par tial derivative<(partial derivative)over bar>S = 0, chi(C)S = 0 and S + C i s (partial derivative + <(partial derivative)over bar>) -exact. If C is a s mooth rational curve, more precise results are given in connection with the normal bundle N-C\M.