PomeLemma 9.11.68.7426C6119D9F Every bounded $U\in \mathrm{PSH}(Y,k)$ satisfies $\int _ Y(k+\mathrm{dd}^{\mathrm{c}}U)^ n=M$. The same holds with $H$ in place of $k$.
Lemma 9.11.687426C6119D9F
Proof. The products are the bounded products of proposition 5.28.2. For $r\geq 1$, the bounded local recursion and Stokes' theorem on the compact manifold $Y$ give
\[ \int _ Y k^{n-r}\wedge (\mathrm{dd}^{\mathrm{c}}U)^ r =\int _ YU\, \mathrm{dd}^{\mathrm{c}}(k^{n-r})\wedge (\mathrm{dd}^{\mathrm{c}}U)^{r-1}=0. \]
Alternatively, one may convolve $U$ on the torus, add a fixed local positive quadratic function, and pass to the limit in the decreasing bounded expansion; every smooth integral vanishes by the same identity. Expanding the top power proves the assertion. The proof for $H$ is identical, since its powers are also $\mathrm{dd}^{\mathrm{c}}$-closed and its mass is $M$. $\square$
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