## Slicing and extension of closed positive currents. (Tranchage et prolongement des courants positifs fermés.)(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.

### MSC:

 32C30 Integration on analytic sets and spaces, currents

### Keywords:

closed positive currents; slicings
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