A balanced proper modification of P3

A proper modification of a Kähler manifold is not necessarily Kähler. The authors prove that, instead, the classical example of Hironaka of a non-Kähler proper modification of P3, biholomorphic outside the curve with equations y2 = x2 +x3 and z = 0 in P3, is balanced. Recall that a compact manifold is balanced if it admits a Hermitian metric whose Kähler form is co-closed. The proof relies on the following technical result: The only positive (p, p) current T satisfying (de-debar)T = 0, whose support has zero Hausdorff 2p-measure, is T = 0.