Lemma 5.29.25CEF121C5391

Proof. Cover $X$ by finitely many coordinate balls $B'_ j$, with larger concentric balls $B_ j$, $B'_ j\Subset B_ j$. A local capacity test $0\leq v\leq 1$ on $B_ j$ can be turned into a global test as follows. Choose a smooth function $\chi _ j$ with $-2\leq \chi _ j\leq 0$, equal to zero near $\overline{B'_ j}$ and to $-2$ near $\partial B_ j$. Choose $\delta >0$ so small that

The function

is $\omega $-plurisubharmonic on $X$, because near $\partial B_ j$ the constant branch is the larger one, and it takes values between zero and one. On $B'_ j$ we have $\omega _ V\geq \delta \mathrm{dd}^{\mathrm{c}}v$. By the positivity of bounded products,

The local volume–capacity estimate, summed over the finite cover, proves (5.29.3).

For the second estimate, choose on each $B_ j$ a smooth negative quadratic function $r_ j$ with $\mathrm{dd}^{\mathrm{c}}r_ j\geq \omega $. For every global test $0\leq v\leq 1$, the functions $u+r_ j\leq 0$ and $v+r_ j$ are plurisubharmonic on $B_ j$. Thus

Here the second inequality uses $\omega _ v\leq \mathrm{dd}^{\mathrm{c}}(v+r_ j)$, and the third is the weighted Chern–Levine–Nirenberg inequality, applied to local test functions with a common bound. The $L^1$ bound for normalized potentials in theorem 5.25.2 completes the proof after taking the supremum over $v$. $\square$