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Plurisubharmonic currents and their extension across analytic subsets. (English) Zbl 0784.32014
Let \(\Omega\) be an open subset of \(\mathbb{C}^ N\) and \(Y\) an analytic subset of \(\Omega\); a current \(T\) on \(\Omega\) is called plurisubharmonic if the current \(i\partial\overline\partial T\) is positive. A classical theorem of Grauert and Remmert asserts that if \(f\) is a plurisubharmonic function on \(\Omega-Y\), locally bounded from above in \(\Omega\), then \(f\) extends to a unique plurisubharmonic function on \(\Omega\).
Our main result is an analogous theorem for currents: If \(T\) is a negative plurisubharmonic current on \(\Omega-Y\) of bidimension \((p,p)\), and \(\dim Y<p\), then \(T\) extends to a negative plurisubharmonic current on \(\Omega\).
For \(p<N\) \((p=N\) is the case of functions) the plurisubharmonic extension is not unique, if we do not pretend that it has measure coefficients; on the contrary, there is a unique extension of order zero, namely the simple extension \(T^ 0\).
Moreover, we get the following geometric characterization: if \(T\) is a negative plurisubharmonic current on \(\Omega-Y\) of bidimension \((p,p)\), and \(\dim Y<p\), then \(i\partial\overline\partial T^ 0- (i\partial\overline\partial T)^ 0=\sum_ ic_ i[Y]_ i\), where \(Y_ i\) are the irreducible components of \(Y\) of dimension \(p\), and \(c_ i\geq 0\).

MSC:
32U05 Plurisubharmonic functions and generalizations
32C30 Integration on analytic sets and spaces, currents
32B15 Analytic subsets of affine space
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