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Principles of Algebraic Geometry
Let A be a closed subset of an open subset Ω of ℂ n and T be a negative current on Ω\ A of bidimension (p,p). Assume that T is psh and A is complete pluripolar such that the Hausdorff measure ${{{{\cal{ H}}}_{{2p}}(\overline{{{{\rm{ Supp}}} T}}\cap A)=0}}$ , then T extends to a negative psh current on Ω. We also show that if T is psh or if dd cT extends to a current with locally finite mass on Ω, then the trivial extension ${{\widetilde{{T}}}}$ of T by zero across A exists in both cases: A is the zero set of a k−convex function with k≤p−1 or ${{{{\cal{ H}}}_{{2(p-1)}}(\overline{{{{\rm{ Supp}}} T}}\cap A)=0}}$ . Our basic tool is the following theorem [El3]: Let A be a closed complete pluripolar subset of an open subset Ω of ℂ n and T be a positive current of bidimension (p,p) on Ω\ A. Suppose that ${{\widetilde{{T}}}}$ and ${{\widetilde{{dd^cT}}}}$ exist (resp. ${{\widetilde{{T}}}}$ exists and dd cT ≤0 on Ω\ A), then there exists a positive (resp. closed positive) current S supported in A such that ${{\widetilde{{dd^cT}}=dd^c\widetilde{{T}}+S}}$ . Furthermore, we give a generalization of some theorems done by Siu and Ben Messaoud-El Mir and Alessandrini-Bassanelli without requiring anything from dT.
Mathematische Zeitschrift – Springer Journals
Published: Sep 4, 2003
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