Proposition 14.22.13.F563CACF5D7D Let $n\geq 2$, and let $u,v$ be smooth $\omega $-plurisubharmonic functions with $A^ n=B^ n=f\, \mathrm{d}V$ for a smooth function $f\geq 0$. Then
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14.22.14
\begin{equation} \int _ X \mathrm{d}w\wedge \mathrm d^{\mathrm c}w\wedge T =\frac12\int _ X w^2\mathrm{dd}^{\mathrm{c}}T. \end{equation}
If $\mathrm{dd}^{\mathrm{c}}T=0$ and $\{ f>0\} $ is dense in $X$, then $u-v$ is constant.
Proof.
By (14.20.1), $\mathrm{dd}^{\mathrm{c}}w\wedge T=0$. Integrating (14.22.5) against $T$, and applying Stokes' theorem twice to the left-hand side, yields
\[ \int _ X\mathrm{dd}^{\mathrm{c}}(w^2)\wedge T=\int _ Xw^2\mathrm{dd}^{\mathrm{c}}T. \]
Substitution proves (14.22.14).
For the uniqueness assertion, apply 14.22.1(1) with $h=1$. At every point where $f>0$, the form $B$ is positive definite, so the term $B^{n-1}$ makes the cofactor $T$ strictly positive. The set where $T$ is strictly positive is therefore dense, as required.
$\square$
Pome
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