Lemma 5.29.17.7EE2FAA017DE Let $\Omega $ be a Hermitian form and $\theta $ a smooth real $(1,1)$-form on $X$. Suppose that $\Omega $-plurisubharmonic functions $v_ j$ have common finite upper and lower bounds and decrease to $v$. Then the signed measures
\[ \nu _ j=(\theta +\mathrm{dd}^{\mathrm{c}}v_ j)^ n \]
converge to $(\theta +\mathrm{dd}^{\mathrm{c}}v)^ n$ against every fixed bounded quasi-continuous function. Their total variations are bounded, uniformly in $j$, by a constant multiple of the local capacity on the balls of a fixed finite cover.
Proof.
On a relatively compact coordinate ball, choose a fixed quadratic function $q$ such that the functions $v_ j+q$ are plurisubharmonic; they still have common bounds. Expand
\[ (\theta +\mathrm{dd}^{\mathrm{c}}v_ j)^ n =\sum _{\ell =0}^ n\binom n\ell \bigl (\mathrm{dd}^{\mathrm{c}}(v_ j+q)\bigr )^\ell \wedge (\theta -\mathrm{dd}^{\mathrm{c}}q)^{n-\ell }. \]
The absolute value of the $\ell $-th summand is bounded by a fixed constant times
\[ \bigl (\mathrm{dd}^{\mathrm{c}}(v_ j+q)\bigr )^\ell \wedge \kappa _0^{n-\ell }, \]
where $\kappa _0$ is a fixed positive Euclidean form. This positive mixed measure is bounded, uniformly in $j$, by a constant times the local capacity: add a fixed strictly plurisubharmonic quadratic function to $v_ j+q$, expand the top power of its Hessian, and rescale its uniformly bounded range into $[0,1]$ to apply the definition of capacity. This also covers $\ell =0$. Summing over a finite cover gives the asserted control of the total variations. The ordinary decreasing continuity of Bedford–Taylor products gives weak convergence term by term. For a bounded quasi-continuous coefficient, remove an open set of small capacity and extend the restriction of the coefficient to the compact complement to a continuous function on $X$. The variation bound controls the resulting error uniformly for the sequence and its limit, and letting the capacity of the removed set tend to zero proves the assertion.
$\square$
Pome
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