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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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