Lemma 7.3.3.9C6450BF0E32 Let $B$ be a ball and $R$ a closed positive current of bidimension $(1,1)$ on a neighborhood of $\overline B$. Let $a$ be a smooth function and $b$ a bounded plurisubharmonic function on a neighborhood of $\overline B$, and suppose that $a=b$ on a collar of $\partial B$. Then
\[ \int _ B a\, \mathrm{dd}^{\mathrm{c}}b\wedge R=\int _ B b\, \mathrm{dd}^{\mathrm{c}}a\wedge R. \]
In particular one may take $R=S\wedge \gamma $, where $S$ is a closed positive current, $\gamma $ is a smooth closed form of pure bidegree, and $S\wedge \gamma $ is positive of bidimension $(1,1)$.
Proof.
Both integrals are finite, since $a$ and $b$ are bounded on $B$ and $\mathrm{dd}^{\mathrm{c}}a\wedge R$ and $\mathrm{dd}^{\mathrm{c}}b\wedge R$ are measures of finite total variation near $\overline B$. The function $h=b-a$ is bounded and Borel, and it vanishes on a collar $V$ of $\partial B$; hence its restriction to $B$ vanishes outside the compact set $K_0=\overline B\setminus V\subset B$. On $B$ we have $bR=aR+hR$, and since $R$ is closed, $\mathrm{dd}^{\mathrm{c}}(aR)=\mathrm{dd}^{\mathrm{c}}a\wedge R$ by (7.2.3); thus
\[ \mathrm{dd}^{\mathrm{c}}b\wedge R=\mathrm{dd}^{\mathrm{c}}a\wedge R+\mathrm{dd}^{\mathrm{c}}(hR)\quad \text{on }B, \]
where $\mathrm{dd}^{\mathrm{c}}(hR)$ is a signed measure with support in $K_0$. Choose $\chi \in C^\infty _ c(B)$ equal to $1$ on a neighborhood of $K_0$. By the definition of $\mathrm{dd}^{\mathrm{c}}$ on currents,
\[ \int _ Ba\, \mathrm{dd}^{\mathrm{c}}(hR)=\int \chi a\, \mathrm{dd}^{\mathrm{c}}(hR)=\int h\, \mathrm{dd}^{\mathrm{c}}(\chi a)\wedge R=\int _ Bh\, \mathrm{dd}^{\mathrm{c}}a\wedge R. \]
Consequently
\[ \int _ Ba\, \mathrm{dd}^{\mathrm{c}}b\wedge R=\int _ Ba\, \mathrm{dd}^{\mathrm{c}}a\wedge R+\int _ Bh\, \mathrm{dd}^{\mathrm{c}}a\wedge R=\int _ Bb\, \mathrm{dd}^{\mathrm{c}}a\wedge R. \]
The last assertion holds because $S\wedge \gamma $ is then closed, by the Leibniz rule, and positive of bidimension $(1,1)$; moreover $\mathrm{dd}^{\mathrm{c}}b\wedge (S\wedge \gamma )=(\mathrm{dd}^{\mathrm{c}}b\wedge S)\wedge \gamma $ by proposition 7.2.2(4).
$\square$
Pome
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