A top-degree product can carry mass on a pole set, and the non-pluripolar product discards this mass. When the unbounded locus is sufficiently small, the full local current product is well defined. Singularity order then compares its residual measures on the pole set, and bounded perturbations leave these measures unchanged.
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.
$\square$
The subscript distinguishes this product from the non-pluripolar product used later; the two coincide for locally bounded functions.
Lemma 5.27.19.F7D352207205 Let $\Omega \subset \mathbb C^ n$ be a domain and $S\subset \Omega $ a closed set of Hausdorff dimension zero. Let $a,b$ be plurisubharmonic on $\Omega $ and locally bounded off $S$, with $\{ a=-\infty \} =S$. If $a\leq b+C$ near $S$ for some finite constant $C$, then, as positive Radon measures,
\[ \mathbf1_ S(\mathrm{dd}^{\mathrm{c}}b)^ n_{\mathrm{full}} \leq \mathbf1_ S(\mathrm{dd}^{\mathrm{c}}a)^ n_{\mathrm{full}}. \]
If $a-b$ is bounded near $S$, equality holds.
Proof.
Fix a compact set $E\subset S$ and $\varepsilon >0$, and put $w_ c=\max \{ b-c,(1+\varepsilon )a\} $ for $c>0$. Since $a=-\infty $ on the compact set $E$, upper semicontinuity gives a neighborhood of $E$ on which $a<-(C+c)/\varepsilon $ and $a\leq b+C$. Off $S$, on this neighborhood, $b-c-(1+\varepsilon )a\geq -C-c-\varepsilon a>0$, while on $S$ the second branch of the maximum equals $-\infty $. Thus $w_ c=b-c$ near $E$, and locality on open sets gives $(\mathrm{dd}^{\mathrm{c}}w_ c)^ n_{\mathrm{full}}(E)=(\mathrm{dd}^{\mathrm{c}}b)^ n_{\mathrm{full}}(E)$.
As $c\to \infty $, the functions $w_ c$ decrease to $(1+\varepsilon )a$. Theorem 5.27.18 applies to every $w_ c$ with the same set $S$, so their full measures converge weakly to $(1+\varepsilon )^ n(\mathrm{dd}^{\mathrm{c}}a)^ n_{\mathrm{full}}$. Since the mass of a compact set is upper semicontinuous under weak convergence of positive measures, we obtain
\[ (\mathrm{dd}^{\mathrm{c}}b)^ n_{\mathrm{full}}(E) \leq (1+\varepsilon )^ n(\mathrm{dd}^{\mathrm{c}}a)^ n_{\mathrm{full}}(E). \]
Here we apply the semicontinuity after multiplying the measures by a fixed nonnegative cutoff function with compact support, equal to one near $E$; both masses on $E$ are finite by the theorem. Letting $\varepsilon \downarrow 0$ and applying inner regularity to all compact sets $E\subset S$ gives the asserted inequality of measures. If $a-b$ is bounded near $S$, then also $\{ b=-\infty \} =S$, so the roles of $a$ and $b$ can be interchanged, and equality follows.
$\square$
Pome
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