Slicing and extension of closed positive currents. (Tranchage et prolongement des courants positifs fermĂ©s.) (French) Zbl 0879.32009

Let \(\Delta^n\) be the unit polydisk in \(\mathbb C^n.\) For \(z=(z_1,\dots,z_n)\in\Delta^n=\Delta^k\times\Delta^{n-k},\quad 1\leqq k\leqq n,\) it is assumed that \(z=(z',z''), \quad \pi(z)=z'=(z_1,\dots,z_k)\) and \(|z|=(\sum_{j=1}^nz_j\bar z_j)^{1/2}.\) The extension theorem for a closed positive current with conditions on the slicings is proved. Let \(A\) be a closed pluripolar complete set in the unit polydisk \(\Delta^n\) and \(T\) be a positive closed current in \(\Delta^n\setminus A\) of complex dimension \(p,\quad k\leqq p<n,\) such that: (i) There exists \(r,\quad 0\leqq r < 1,\) for which \(T\) is of finite mass in the neighbourhood of points of \(\{(z',z'')\in\Delta^n/r<|z''|\}.\) (ii) There exists a non-pluripolar set \(F\subset\Delta^k\) of finite mass such that for all \(a\in F,\) the slicing \(\langle T,\pi,a\rangle\) exists on \(\Delta^n\setminus A.\) Then, the trivial extension of \(T\) by zero over \(A\) is a closed positive current.


32C30 Integration on analytic sets and spaces, currents
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