PomeD36A46B4DD86Lemma 13.14.2.D36A46B4DD86 With the notation above,
More generally, let $\theta _1,\ldots ,\theta _ n$ be smooth real $(1,1)$-forms, let $u^ i\in \mathrm{PSH}(X,\theta _ i)$, and let $D$ be the set where $u^ i>-t$ for every $i$. Then the mixed product of the currents $\theta _ i+\mathrm{dd}^{\mathrm{c}}\max (u^ i,-t)$ is positive on $D$.
Proof. Work in a coordinate ball with a fixed positive constant form $\kappa $, and fix $\epsilon >0$. Every point has a smaller neighborhood carrying a constant real form $\theta '$ such that
Take a smooth local potential $r$ for $\theta '+\epsilon \kappa $. Then $u+r$ is plurisubharmonic. On a relatively compact smaller neighborhood choose $k>t+\sup |r|$ and set
This function is locally bounded and equals $u_ t=u$ on $D_ t$. Moreover,
is the sum of a positive closed current with bounded potential and a smooth positive form, so its bounded products are positive. By plurifine locality, the restriction of $(\theta +\mathrm{dd}^{\mathrm{c}}u_ t+2\epsilon \kappa )^ n$ to $D_ t$ is positive on this neighborhood.
These neighborhoods cover the coordinate ball. Now let $\epsilon \downarrow 0$. The product with the fixed bounded potential $u_ t$ is a polynomial in $\epsilon $ whose coefficients have locally finite variation, so it converges in local total variation, also after restriction to $D_ t$. This proves the first inequality. The same construction, applied to each factor, gives the positivity of mixed products.
For the second inequality, expand the difference of the two powers by the telescoping identity, using the smooth positive form $\Omega -\theta $. Every summand is a mixed product of factors that are positive on $D_ t$, as just proved. The difference is therefore nonnegative there. $\square$
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