Compact complex manifolds which are Kähler outside an analytic subset

This lecture announces results concerning compact complex manifolds M which are Kähler outside an analytic subset Y of codimension at least 2 (the authors conjecture that such a manifold has always a Kähler modification). The question is to determine to what extent the “exceptional” subset Y is an obstruction for M to be Kähler. The results are restricted to the case when Y is a curve. The first theorem asserts that the (n−1, n−1)-current defined by Y always yields to an obstruction in the spirit of Harvey-Lawson’s criterion when M is not Kähler. The authors also obtain numerical conditions.