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Complete locally pluripolar sets. (English) Zbl 0711.32008
Let X be a complex space and \(A\subset X\) a closed subset. A is called complete locally pluripolar if for any \(x_ 0\in A\) there is an open neighbourhood \(U=U(x_ 0)\) and a plurisubharmonic function \(\phi: U\to [-\infty,\infty)\) such that \(A\cap U=\{\phi =-\infty \}.\)
We show that in a Stein space X any complete locally pluripolar set A is globally complete pluripolar, i.e. there exists a plurisubharmonic function \(\psi\) on X such that \(A=\{\psi =-\infty \}\). A result of this type for locally pluripolar sets, i.e. locally contained in \(\{\phi =- \infty \}\), had been proved long ago by Josefson.
When X is q-complete we show that the global defining function \(\psi\) may be chosen smooth and strongly q-convex outside A. From this last statement we deduce that any q-complete subset which is complete locally pluripolar has a q-complete open neighbourhood. In particular any q- complete closed analytic subset of a complex space has a q-complete open neighbourhood, which generalizes a well-known result of Siu.
Reviewer: Mihnea Colţoiu

MSC:
32U05 Plurisubharmonic functions and generalizations
32F10 \(q\)-convexity, \(q\)-concavity
32C25 Analytic subsets and submanifolds
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