Continuation of positive, closed currents and analytic sets
Let \(\Omega\) be an open subset of \(\mathbb{C}^ n\) and let \(u:\Omega\to\mathbb{R}^ +_ 0\) be a plurisubharmonic function with the following properties: (i) \(dd^ cu\geq\lambda_ 0\sum^ n_{j=k+1}idz_ j\wedge d\overline z_ j\) for some \(\lambda_ 0\in\mathbb{R}^ +\) (in distributional sense) (ii) There is an \(\alpha\in]0,1[\) such that \(u^{1-\alpha}\) is still plurisubharmonic. Let \(A\) be a closed subset of \(u^{-1}(\{0\})\). If \(T\) is a closed, positive current of bidimension \((p,p)\) on \(\Omega\backslash A\), \(p>k+{1\over 2\alpha}\), then the trivial extension \(\tilde T\) of \(T\) to \(\Omega\) is the unique closed, positive extension of \(T\) to \(\Omega\). A proof of this theorem is given and using slicing-theory this result is generalized to the zero-set of a non-negative, strictly \(q\)-convex function \(u\). If \((\Omega,ds^ 2)\) is a Kähler manifold the following theorem is obtained: Let \(A\subset\Omega\) be closed and let \(X\subset\Omega\backslash A\) be an analytic subset with locally finite volume near \(A\). If there exists a continuous, subharmonic function \(u:\text{Reg} X\to\mathbb{R}\), which is bounded from above and for which \(\lim_{{x\to a\atop x\in\text{Reg} X}}u(x)=-\infty\) \(\forall a\in A\cap\overline X\), then \(\overline X\) is an analytic continuation of \(X\) to \(\Omega\).
scientific article; zbMATH DE number 561293