Proposition 13.14.21.D889AEAF2866 Suppose $v_+(\omega )=+\infty $. For the potential $U$ and its smooth decreasing approximants $U_ N$ constructed in the proof of theorem 13.14.18, the current $T=\omega +\mathrm{dd}^{\mathrm{c}}U$ has integrable coefficients and
\[ \| \omega +\mathrm{dd}^{\mathrm{c}}U_ N-T\| _{L^1(\gamma )} \leq 2^{1-N}M. \]
Its pointwise determinant satisfies
\[ \mathrm{MA}^{\mathrm{ext}}_\omega (U) \geq (\det _\gamma T)\gamma ^ n, \qquad \int _ X(\det _\gamma T)\gamma ^ n=+\infty . \]
Moreover,
\[ \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}U_ N)^ n\longrightarrow +\infty . \]
Proof.
Write $T_ j=\omega +\mathrm{dd}^{\mathrm{c}}u_ j$ for the smooth positive forms in that construction, and put
\[ S_ N=\sum _{j\leq N}a_ jT_ j,\qquad \delta _ N=\sum _{j>N}a_ j=2^{-N}. \]
The smooth positive forms $S_ N$ increase pointwise, and the Gauduchon identity gives
\[ \int _ XS_ N\wedge \gamma ^{n-1}=(1-\delta _ N)M. \]
By monotone convergence of their traces, the series of matrices has a finite positive limit $S$ almost everywhere. The coefficients of $S$ are integrable, because in a $\gamma $-unitary frame the absolute value of every coefficient of a positive matrix is bounded by its trace. Positivity and the same trace identity give
\[ \| S-S_ N\| _{L^1(\gamma )}=\delta _ N M. \]
In particular, $S_ N\to S$ in the sense of distributions. Since
\[ \omega +\mathrm{dd}^{\mathrm{c}}U_ N=S_ N+\delta _ N\omega \longrightarrow \omega +\mathrm{dd}^{\mathrm{c}}U \]
in the sense of distributions, by $L^1$ convergence of the potentials, we obtain $T=S$. The triangle inequality now gives
\[ \| \omega +\mathrm{dd}^{\mathrm{c}}U_ N-T\| _{L^1(\gamma )} \leq \| S_ N-S\| _{L^1(\gamma )} +\delta _ N\| \omega \| _{L^1(\gamma )} =2\delta _ N M. \]
To compare $\mathrm{MA}^{\mathrm{ext}}_\omega (U)$ with the determinant, fix $N$ and set $V_ N=U-U_ N$. The remaining partial sums give $\delta _ N\omega +\mathrm{dd}^{\mathrm{c}}V_ N\geq 0$ and $V_ N\leq 0$. For $t>0$, the function $W_{N,t}=U_ N+\max (V_ N,-t)$ is bounded, and its current is the sum of two positive terms:
\[ \omega +\mathrm{dd}^{\mathrm{c}}W_{N,t} =S_ N+\bigl (\delta _ N\omega +\mathrm{dd}^{\mathrm{c}}\max (V_ N,-t)\bigr ). \]
On $D_ t=\{ U>-t\} $, the inequality $U_ N\leq 0$ gives $V_ N\geq U>-t$, and consequently $W_{N,t}=U=\max (U,-t)$ there. Locality and positivity of bounded products imply
\[ \mathbf1_{D_ t}\mathrm{MA}^{\mathrm{ext}}_\omega (U) \geq \mathbf1_{D_ t}S_ N^ n. \]
The smooth measure $S_ N^ n$ gives no mass to the pluripolar pole set of $U$, so letting $t\to \infty $ proves $\mathrm{MA}^{\mathrm{ext}}_\omega (U)\geq S_ N^ n$. The positive matrices $S_ N$ increase to $T$ almost everywhere, so their determinants increase to $\det _\gamma T$. Monotone convergence on every Borel set gives the asserted inequality between $\mathrm{MA}^{\mathrm{ext}}_\omega (U)$ and the determinant measure.
Finally, for every $N\geq j$, positivity gives $S_ N\geq a_ jT_ j$. Hence both the measurable limit and the smooth approximating forms satisfy
\[ \int _ X(\det _\gamma T)\gamma ^ n \geq a_ j^ n\int _ XT_ j^ n\geq 2^ j, \qquad \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}U_ N)^ n \geq a_ j^ n\int _ XT_ j^ n\geq 2^ j. \]
The first bound holds for every $j$, and the second for every $N\geq j$; together they prove the two assertions about infinite mass.
$\square$
Pome
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