Example 8.10.7AEB4F2397BBF

Proof. Choose holomorphic coordinates $z_ j=x_ j+\mathrm i y_ j$ and a smooth nonnegative cutoff $\chi $ supported in the chart and equal to one on a smaller coordinate box. Extend

by zero outside the chart. These functions are globally continuous and bounded. To check their curvature, replace $|x_ j|$ by $g_\delta (x_ j)=(x_ j^2+\delta ^2)^{1/2}$, with $0<\delta \leq 1$. The Hessian of $\chi g_\delta $ has the nonnegative term $\chi \mathrm{dd}^{\mathrm{c}}g_\delta $; all its remaining terms are uniformly bounded forms, since $g_\delta $ and $g_\delta '$ are uniformly bounded on the fixed support. Thus $\mathrm{dd}^{\mathrm{c}}(\chi g_\delta )\geq -C\omega $ with $C$ independent of $\delta $. Choosing $a>0$ sufficiently small gives $\omega +\mathrm{dd}^{\mathrm{c}}(a\chi g_\delta )\geq \omega /2$ for both coordinates. Their decreasing limits $u,v$ are therefore $\omega $-plurisubharmonic and admissible.

On the box where $\chi =1$, set $\alpha _ j=\frac{\mathrm i}{4\pi }\mathrm dz_ j\wedge \mathrm d\bar z_ j$. The book's normalization gives

The measures $g_\delta ^{\prime \prime }(t)\, \mathrm dt$ have mass two, while their mass outside any fixed neighborhood of zero tends to zero. The two smooth factors depend on distinct real variables, so their product is an approximate identity in $(x_1,x_2)$ and converges to the tensor-product measure. By 8.10.5, locally

Choose $E$ to be a compact piece of $\{ x_1=x_2=0\} $ inside this box with nonempty relative interior. The preceding lemma gives $\operatorname {Cap}_1(E)=0$. On this box, the positive measure $D_2(u,v)\wedge \omega ^{n-2}$ has strictly positive mass on $E$, since $\omega $ is strictly positive. The smooth term $\omega ^ n$ and the two single-Hessian terms in the expansion

have zero mass on $E$: each single-Hessian term is absolutely continuous in the other normal real variable. Hence the asserted mass is strictly positive. A bound by $C\operatorname {Cap}_1(F)^ b$ for all Borel $F$, with any $b>0$, would contradict this choice of $E$. $\square$