Theorem 5.27.18`1BBBF8109419`

Theorem 5.27.18.1BBBF8109419 Let $\Omega \subset \mathbb C^ n$ be a domain and $S\subset \Omega $ a closed set of Hausdorff dimension zero. Suppose that $u_1,\ldots ,u_ q$, $1\leq q\leq n$, are plurisubharmonic on $\Omega $ and locally bounded on $\Omega \setminus S$. Then their inductively defined full current product

\[ \bigl (\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\bigr )_{\mathrm{full}} \]

is a closed positive current of order zero. It is symmetric, multilinear, local in the ordinary topology, and continuous under simultaneous decreasing approximation by plurisubharmonic functions. The weighted current $u_1\bigl (\mathrm{dd}^{\mathrm{c}}u_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\bigr )_{\mathrm{full}}$ has coefficients of locally finite variation. These full products may charge $S$.

Proof. Apply the classical theorem on products with unbounded potentials, together with its monotone continuity statement, in [Chapter III, Theorem 4.5 and Proposition 4.9, pp. 152–156, DemCADG], with the underlying current $T=1$. For every choice of $m$ of the functions, that theorem requires the intersection of their unbounded loci to have zero Hausdorff measure of dimension $2n-2m+1$. This intersection lies in $S$, and $2n-2m+1\geq 1$, so the measure is zero. The same hypothesis gives local integrability of the weighted current. Symmetry, multilinearity and locality on open sets follow from the local construction of the currents and decreasing regularization. $\square$

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@misc{pome-tag-1BBBF8109419,
  author = {{Pome contributors}},
  title = {Theorem 5.27.18},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/1BBBF8109419/}
}

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Theorem 5.27.181BBBF8109419

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Theorem 5.27.18`1BBBF8109419`