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Courants kählériens et surfaces compactes. (Kähler currents and compact surfaces). (French) Zbl 0926.32026
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 [K. Kodaira and J. Morrow, ‘Complex manifolds’, New York: Holt, Rinehart and Winston (1971)], and the solution was received by 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 \(X.\)

32J15 Compact complex surfaces
32C30 Integration on analytic sets and spaces, currents
Full Text: DOI Numdam EuDML
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