Example 13.22.8AB2E8992474C

Proof. Fix a point $x$, and let $g_ x\leq 0$ be the Green function used in theorem 20.3.10, with $\mathrm{dd}^{\mathrm{c}}g_ x=\delta _ x-\theta $, where $\delta _ x$ is the Dirac measure at $x$. For $0\leq a<1$, put $q_ a=P_\theta [ag_ x]$. The current $\theta +\mathrm{dd}^{\mathrm{c}}(ag_ x)$ equals $a\delta _ x+(1-a)\theta $, with non-pluripolar mass $1-a$. Since the background is closed, both singularity volumes of $ag_ x$ equal this mass, and theorem 13.19.5 gives

Since $g_ x\leq 0$, the potentials $q_ a$ decrease as $a$ increases, and (13.22.2) gives $d_\theta (q_ a,q_ b)=|a-b|$. Choose $a_ j<1-\delta $ with $a_ j\uparrow 1-\delta $. The sequence $q_{a_ j}$ is $d_\theta $-Cauchy, and every term has mass $1-a_ j>\delta $. By (13.22.2), any $d_\theta $-limit has upper volume $\lim _ j(1-a_ j)=\delta $. In this closed background the two volumes coincide, so no such limit has mass strictly greater than $\delta $. $\square$