Generalized Kähler metrics and proper modifications (Q1769144)

By \textit{B. G. Moishezon} [Singular Kählerian spaces, Manifolds, Proc. int. Conf. Manifolds relat. Top. Topol., Tokyo 1973, 343--351 (1975; Zbl 0344.32018)] a reduced complex space \(X\) is Kähler if there is an open covering \(\{U_\alpha\}_\alpha\) of \(X\) and smooth strictly plurisubharmonic functions \(\varphi_\alpha\) on \(U_\alpha\) such that \(\varphi_\alpha-\varphi_\beta\) equals locally the real part of holomorphic functions. The author extends this to non-necessarily smooth functions \(\varphi_\alpha\); one obtains ``generalized Kähler spaces''. The result is: Theorem 2.5. Let \(p: X\to Y\) be a proper modification of reduced complex spaces. Suppose that \(X\), \(Y\) are compact and \(p\) is surjective. If \(Y\) is Kähler, then \(X\) is generalized Kähler.