Abstract

The authors prove that a subset \(S\) of a compact Kähler manifold \(X\) with fundamental form \(\omega\) is locally pluripolar if and only if there exists \(\varphi\in {\mathcal E}^\infty(X,\omega)\) such that \(\varphi\not\equiv -\infty\) and \(S\subset \varphi^{-1}(-\infty)\), where \({\mathcal E}^\infty(X,\omega)=\bigcap_{p\geq 1}{\mathcal E}^p(X,\omega)\) and \({\mathcal E}^p(X,\omega)\) consists of all \(\varphi\in \text{PSH}(X,\omega)\) such that there exists a sequence \(\varphi_j\in \text{PSH}(X,\omega)\cap L^\infty(X)\) with \(\varphi_j\searrow \varphi\) and \(\sup_{j\geq 1}\int_X| \varphi_j| ^p (\omega+dd^c\varphi_j)^n<\infty\). They prove that a subset \(S\) of \({\mathbb C}{\mathbb P}^n\) is complete \(\omega_{FS}\)-pluripolar, where \(\omega_{FS}\) denotes the Fubini-Study form, if and only if all its intersections with the fundamental coordinate neighbourhoods are complete pluripolar if and only if its inverse image under the natural projection is complete pluripolar in \({\mathbb C}^{n+1}\). Finally, they investigate the existence of \(\omega\)-subextentions of psh functions defined in some hyperconvex domain in \(X\).