Proposition 15.26.28F3C95237950

Proof. In the coordinate ball write $R=|z|$ and set

extending $\psi _ j$ by one outside it. On the transition region of $\chi $, $R$ is bounded away from zero and exceeds all $r_ j$. Hence all $\psi _ j$ agree there with the same smooth function. Where $\chi =1$ they are plurisubharmonic, and outside its support they are constant. One fixed $K\geq 1$ therefore satisfies $\mathrm{dd}^{\mathrm{c}}\psi _ j\geq -Kh$ for every $j$. Put

The estimate $\| \psi _ j\| _\infty \leq C(1+|\log r_ j|)$ gives uniform convergence, and $\sum d_ j=1/(2K)$ yields $h+\mathrm{dd}^{\mathrm{c}}u_0\geq h/2$.

For two points of the inner ball at Euclidean distance $t$, the reverse triangle inequality gives

The contribution of $r_ j\geq t$ is at most $Ct\sum _{r_ j\geq t}r_ j^{-1/2}\leq C\sqrt t$. For $r_ j<t$, summing the geometric tail gives at most $C\sqrt t\sum _{\ell \geq 0}2^{-\ell /2}(1+\ell )$. On the cutoff transition and outside, the series is a fixed smooth function. These estimates give a local $1/2$-Hölder bound near every point. A finite cover and the bound on the oscillation give the global bound.

On the smaller ball, the currents $\mathrm{dd}^{\mathrm{c}}\psi _ j$ are closed and positive. Expand the product of $h+\sum _{j\leq N}d_ j\mathrm{dd}^{\mathrm{c}}\psi _ j$ there and retain its positive diagonal terms. By lemma 5.27.26, these are $d_ j^ n\sigma _ j$. Uniform convergence of the corrected bounded potentials gives weak convergence of their products. Testing against nonnegative functions and first retaining any fixed finite number of diagonal terms therefore gives

Here the right-hand side is supported strictly inside the smaller ball, so the inequality extends to $X$. Each $\sigma _ j$ is non-pluripolar by the same lemma; countable additivity gives that property for $\rho $. The countable union of the spheres has zero smooth volume, which proves singularity. Local bounded-product estimates on a finite cover make the mass of $u_0$ finite, and the retained inequality makes it positive. Set $k=S_ n/Z^ n$ and subtract $\sup _ Xu_0$ to obtain $u_\rho $. $\square$