Courants kählériens et surfaces compactes. (Kähler currents and compact surfaces) (Q1288636)

The unified proof of the existence of Kähler metric on a surface of even Betti first number is given. The problem was set up by Kodaira [\textit{K. Kodaira} and \textit{J. Morrow}, `Complex manifolds', New York: Holt, Rinehart and Winston (1971)], and the solution was received by \textit{Y.-T.~Siu} [Invent. Math. 73, 139-150 (1983; Zbl 0557.32004)]. The author uses the Demailly regularization theorem and the Siu decomposition theorem to prove that the compact complex manifold \(X\) having a Kähler current is the Kähler manifold outside of some analytic subset of codimension at least two. Using the Hodge symmetry, which follows from the hypothesis \(b_1=2 h^{0,1},\) and the criterion similar to that of Harvey-Lawson, the author proves elementary the existence of a Kähler current on \(X\), that, due to above-stated result, is enough to ensure the existence of the Kähler metric on \(X.\)