Corollary 13.8.8.A8BA63307F6C Let $u_ j,v_ j\in \mathrm{PSH}(X,\theta _ j)$ for $1\leq j\leq p\leq n$, and put $T_ j=\theta _ j+\mathrm{dd}^{\mathrm{c}}u_ j$ and $T'_ j=\theta _ j+\mathrm{dd}^{\mathrm{c}}v_ j$. If every difference $u_ j-v_ j$ is bounded, then
\[ \int _ X\langle T_1\wedge \cdots \wedge T_ p\rangle \wedge \omega ^{n-p} =\int _ X\langle T'_1\wedge \cdots \wedge T'_ p\rangle \wedge \omega ^{n-p}. \]
If only $u_ j\geq v_ j-O(1)$ for every $j$, then the left side is at least the right side.
Proof.
Bounded differences. For real numbers $a_0,a_1,\ldots ,a_ p>0$, apply theorem 13.8.7 in both directions to the potentials $\sum _ ja_ ju_ j$ and $\sum _ ja_ jv_ j$, whose difference is bounded, with respect to the background $a_0\omega +\sum _ ja_ j\theta _ j$. The two resulting masses are equal. By multilinearity, both are homogeneous polynomials of degree $n$ in $(a_0,\ldots ,a_ p)$, and they agree on an open set; fixing all but one variable and using the uniqueness of polynomials in one variable successively, we see that their coefficients agree. By proposition 13.4.3, the coefficient of $a_0^{n-p}a_1\cdots a_ p$ is $n!/(n-p)!$ times the mixed mass in the statement, which proves equality.
The ordered case. Put $v_{j,k}=\max \{ u_ j-k,v_ j\} $. Since $v_ j\leq u_ j+C_ j$ for constants $C_ j$, we have $u_ j-k\leq v_{j,k}\leq u_ j+C_ j$, so each tuple $(v_{j,k})_ j$ has the same mixed mass as $(u_ j)_ j$, by the first part. As $k\to \infty $, $v_{j,k}$ decreases to $v_ j$ and therefore converges to it in extended capacity (lemma 13.7.3). The lower inequality of theorem 13.7.5, with test form $\omega ^{n-p}$, bounds the mixed mass of the $v_ j$ by these equal masses.
$\square$
Pome
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