Pome600C2F850586Let $\theta $ be a smooth real $(1,1)$-form and $u\in \mathrm{PSH}(X,\theta )$. For $t>0$, put
Choose a Hermitian form $\Omega \geq \theta $. Then $u_ t$ is bounded and $\Omega $-plurisubharmonic, so proposition 5.28.2 defines the product $(\theta +\mathrm{dd}^{\mathrm{c}}u_ t)^ n$ as a signed measure. It need not be positive on $X$, but its restriction to the plurifinely open set $D_ t$ is positive. The proof replaces $u_ t$ near each point by a local truncation that preserves positivity up to an arbitrarily small smooth error.
Lemma 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. $\square$
Theorem 13.14.3 (Extended non-pluripolar measure).99C7ACBB2589 Let $\theta $ be a smooth real $(1,1)$-form and $u\in \mathrm{PSH}(X,\theta )$. The measures $\mathbf1_{D_ t}(\theta +\mathrm{dd}^{\mathrm{c}}u_ t)^ n$ increase with $t$. Their limit
E6AA45ACEA33
13.14.4
is a positive Borel measure that puts no mass on pluripolar sets; it is finite on each $D_ t$ and $\sigma $-finite on $X$. It depends only on the current $\theta +\mathrm{dd}^{\mathrm{c}}u$; in particular, it is unchanged when a constant is added to $u$. If $\Omega $ is a Hermitian form with $\theta \leq \Omega $, then
The total mass of $\mathrm{MA}^{\mathrm{ext}}_\theta (u)$ may be infinite.
Proof. For $s>t$, plurifine locality on $D_ t\subset D_ s$ gives
Together with lemma 13.14.2, this proves that the measures increase. Their increasing limit is countably additive by monotone convergence, and it vanishes on pluripolar sets because each bounded product does. The comparison with $\Omega $ passes to the limit. By the compatibility just proved,
and the mass of this measure is bounded by that of the bounded product $(\Omega +\mathrm{dd}^{\mathrm{c}}u_ t)^ n$. The sets $D_ t$ exhaust the complement of $\{ u=-\infty \} $, which has zero mass, so the limit is $\sigma $-finite.
For a smooth real function $f$, consider $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}f$ and $u'=u-f$. On a common superlevel set of $u$ and $u'$, sufficiently deep truncations of $u$ and $u'$ define the same current with bounded potentials, so their restricted products agree. These sets exhaust the complement of the common pole set, which proves invariance under this change of decomposition. By the elliptic-regularity argument in the proof of proposition 5.28.2, any two smooth decompositions of the same current differ in this way, and adding a constant is a special case. $\square$
We call $\mathrm{MA}^{\mathrm{ext}}_\theta (u)$ the extended non-pluripolar Monge–Ampère measure of $u$; the word extended indicates that its total mass may be infinite. On compact $X$ it defines a current of order zero exactly when its total mass is finite. In that case it is the non-pluripolar Monge–Ampère measure in the usual sense, namely the extension by zero across $\{ u=-\infty \} $ of the plurifinely local bounded products. If $u$ is bounded, then $D_ t=X$ for large $t$, and $\mathrm{MA}^{\mathrm{ext}}_\theta (u)=\mathrm{MA}_\theta (u)$. In accordance with the usual notation for non-pluripolar products, we also write $\langle (\theta +\mathrm{dd}^{\mathrm{c}}u)^ n\rangle $ for $\mathrm{MA}^{\mathrm{ext}}_\theta (u)$; lower powers $\langle (\theta +\mathrm{dd}^{\mathrm{c}}u)^ k\rangle $ are defined by the same truncation where they are used.
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