Lemma 8.13.1.69692CE9985B For every bounded admissible list,
B8626F362B85
8.13.2
\begin{equation} \mathcal M_ m(t_1,\ldots ,t_ m) =\int _ X\omega ^ n+ \sum _{\varnothing \ne I\subset \{ 1,\ldots ,m\} } \int _ X t_{i(I)}D_{I^-}\wedge \mathrm{dd}^{\mathrm{c}}(\omega ^{n-|I|}). \end{equation}
Every integral on the right is an integral of bounded canonical values against signed measure coefficients.
Proof.
Expand $Q_ m$ using (8.9.5). The term for $I$ pairs $D_ I$ with the smooth form $\omega ^{n-|I|}$. By (8.9.2),
\[ \langle D_ I,\omega ^{n-|I|}\rangle =\langle t_{i(I)}D_{I^-},\mathrm{dd}^{\mathrm{c}}(\omega ^{n-|I|})\rangle . \]
Since $|I^-|\leq m-1$, 8.9.3 gives measure coefficients for $D_{I^-}$; their bounded product with $t_{i(I)}$ is well defined. Thus the displayed pairing is exactly the integral in the sum, and is finite on compact $X$. The empty subset contributes $\int _ X\omega ^ n$.
$\square$
Pome
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