On the extension of quasiplurisubharmonic functions (Q2151135)

Let \((V,\omega)\) be a compact Kähler manifold and let \(X\) be an analytic subvariety of \(V\). The main result of the paper is the following extension theorem for plurisubharmonic functions stating that if \(\{\omega\}\) belongs to the real Neron-Severi space \(N\!S_{\mathbb{R}}(V)\), then \(PSH(V,\omega)\vert_X=PSH(X,\omega\vert_X)\). The proof depends on the following result. Assume that \((V, \omega)\) admits an adapted Zariski-open Stein cover, i.e. \(V\) can be covered by finitely many open Stein sets \(V_j\), \(j=1,\dots,N\), that \(V\setminus V_j\) is an analytic subvariety of \(V\), and there exists a smooth exhaustion function \(\rho_j\geq0\) on \(V_j\) such that \(\omega=dd^c\rho_j\), \(j=1,\dots,N\). Then, if \(\varphi\in PSH(X,\omega\vert_X)\) then for each \(a>0\), there exists a \(\psi\in PSH(V,\omega)\) such that \(\psi\vert_X=\varphi\) and \(\max_V\psi<\max_X\varphi+a\).