Hypersurface convexity and extension of Kähler forms

The following generalization of a result of Nemirovski (Russ Math. Surv 63(2):381–382, 2008) is proved: if X is either a projective or a Stein manifold and \(K\subset X\) $K\subset X$ is a compact sublevel set of a strictly plurisubharmonic function \(\varphi \) $\phi$ defined in a neighborhood of K, then \(X{\setminus } K\) $X\backslash K$ is a union of positive divisors if and only if \(dd^c\varphi \) $d{d}^c\phi$ extends to a Hodge form on X. For an arbitrary compact subset \(K\subsetneq X\) $K⊊X$ , this gives that \(X{\setminus } K\) $X\backslash K$ is a union of positive divisors if and only if K admits a neighbourhood basis of sublevel sets of strictly plurisubharmonic functions with the \(dd^c\) $d{d}^c$ -extension property.