Let M be a compact complex manifold and C in M be an irreducible curve such that M − C is Kähler. The paper studies the following problem: What are the cohomological obstructions on C so that M is Kähler? This problem is motivated by the fact that if M is a Moishezon manifold, then there exists an analytic subset Y of M with codimension n ≥ 2 such thatM −Y is Kähler. When dimM ≥ 3, the authors show that one and only one of the following cases can occur: (i) M is Kähler, (ii) C is the (1, 1)-component of a boundary, or (iii) C is part of the (1, 1)-component of a boundary. As a consequence, it is shown that if C is smooth with genus g satisfying NC|M · C > (dimM −1)(g −1), then M is Kähler. When M is a threefold, more precise results are given.