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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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