Extension of plurisubharmonic currents

摘要

Let A be a closed subset of an open subset Omega of C-n and T be a negative current on OmegaA of bidimension (p,p). Assume that T is psh and A is complete pluripolar such that the Hausdorff measure H2p((SuppT) over bar boolean AND A), then T extends to a negative psh current on Omega. We also show that if T is psh or if dd(cT) extends to a current with locally finite mass on Omega, then the trivial extension (T) over tilde of T by zero across A exists in both cases: A is the zero set of a k-convex function with kless than or equal top-1 or H2(p-1)((SuppT) over bar boolean AND A) = 0. Our basic tool is the following theorem [El3]: Let A be a closed complete pluripolar subset of an open subset Omega of C-n and T be a positive current of bidimension (p,p) on OmegaA. Suppose that (T) over tilde and (dd(c)T) over tilde exist (resp. (T) over tilde exists and dd(c)T less than or equal to 0 OmegaA), then there exists a positive (resp. closed positive) current S supported in A such that (dd(c)T) over tilde = dd(c)(T) over tilde + 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. [References: 25]