Proposition 16.27.187EC4B0FF8858

Proof. Let $v$ be a competitor for $\operatorname {Cap}_\omega $, so that $v\in \mathrm{PSH}(X,\omega )$ and $-1\leq v\leq 0$. Fix $\alpha $ and write $B'_\alpha =B(0,R_\alpha )$ in its coordinates. Since $\overline{B'_\alpha }$ is a compact subset of a chart, there is $A_\alpha >0$ with $\omega \leq A_\alpha \mathrm{dd}^{\mathrm{c}}|z|^2$ on $B'_\alpha $ (lemma 6.14.1). Then $\mathrm{dd}^{\mathrm{c}}(v+A_\alpha |z|^2)=(\omega +\mathrm{dd}^{\mathrm{c}}v)+(A_\alpha \mathrm{dd}^{\mathrm{c}}|z|^2-\omega )\geq 0$, so $v+A_\alpha |z|^2$ is plurisubharmonic on $B'_\alpha $, by lemma 6.14.4(2), applied on $B'_\alpha $ to the form $A_\alpha \mathrm{dd}^{\mathrm{c}}|z|^2=\mathrm{dd}^{\mathrm{c}}(A_\alpha |z|^2)$, and the multilinearity and positivity of Bedford–Taylor products give $(\mathrm{dd}^{\mathrm{c}}(v+A_\alpha |z|^2))^ n\geq (\omega +\mathrm{dd}^{\mathrm{c}}v)^ n$ on $B'_\alpha $. Put $c_\alpha =1+A_\alpha R_\alpha ^2$. The function $w=(v+A_\alpha |z|^2+1)/c_\alpha $ is plurisubharmonic on $B'_\alpha $ with values in $[0,1]$, and $(\omega +\mathrm{dd}^{\mathrm{c}}v)^ n\leq c_\alpha ^ n(\mathrm{dd}^{\mathrm{c}}w)^ n$ there. Hence $\mu _ v(E)\leq c_\alpha ^ n\operatorname {Cap}_{B'_\alpha }(E)$ for every Borel set $E\subset B'_\alpha $, and taking the supremum over $v$ gives $\operatorname {Cap}_\omega (E)\leq c_\alpha ^ n\operatorname {Cap}_{B'_\alpha }(E)$. Applying this to the sets $E\cap B_\alpha $ and using the subadditivity of $\operatorname {Cap}_\omega $ (proposition 16.25.6), we obtain the upper bound.

For the lower bound fix $\alpha $ and write $B_\alpha =B(0,r)$ and $B'_\alpha =B(0,R)$ in the coordinates, with $r<R$. Choose $L>0$ with $L(R^2-r^2)>1$, and then $b>0$ so small that $\omega \geq bL\mathrm{dd}^{\mathrm{c}}|z|^2$ on $B(0,R)$ (lemma 6.14.1) and that we can choose a number $a$ with

Let $w$ be plurisubharmonic on $B(0,R)$ with $-1\leq w\leq 0$, and put $q=bw-bL|z|^2$. Then $\omega +\mathrm{dd}^{\mathrm{c}}q\geq b\mathrm{dd}^{\mathrm{c}}w\geq 0$ on $B(0,R)$. At the points of $B(0,R)$ where $|z|^2>a/(bL)$, which include a neighborhood of the boundary sphere because $a<bLR^2$, we have $q\leq -bL|z|^2<-a$.

Define $v$ to be $\max \{ q,-a\} $ on $B(0,R)$ and $-a$ elsewhere. Then $v=-a$ on the open set $U$ formed by $X\setminus \overline{B(0,R)}$ and the points of the chart where $|z|^2>a/(bL)$. Since $U$ and $B(0,R)$ cover $X$, and $v$ is $\omega $-plurisubharmonic on each of them, $v\in \mathrm{PSH}(X,\omega )$. It takes values in $[-1,0]$, and on $B(0,r)$ we have $q\geq -b(1+Lr^2)>-a$, so $v=q$ there. Since $\omega +\mathrm{dd}^{\mathrm{c}}q$ is the sum of the semipositive form $\omega -bL\mathrm{dd}^{\mathrm{c}}|z|^2$ and $b\mathrm{dd}^{\mathrm{c}}w$, positivity of the mixed terms gives $(\omega +\mathrm{dd}^{\mathrm{c}}v)^ n\geq b^ n(\mathrm{dd}^{\mathrm{c}}w)^ n$ on $B(0,r)$. The capacity $\operatorname {Cap}_{B(0,R)}$ may equally be computed with competitors taking values in $[-1,0]$, so it follows that

Summing over the finitely many balls gives the lower bound.

It remains to prove (1)–(3). Let $F$ and $G$ be the functions of theorem 16.27.17 for $\operatorname {Cap}_{\theta ,\psi }$; since they are continuous and vanish at zero, a sequence of Borel sets has capacities $\operatorname {Cap}_{\theta ,\psi }$ tending to zero if and only if their capacities $\operatorname {Cap}_\omega $ tend to zero.

(1) Pluripolar Borel sets have capacity zero by proposition 16.25.6. Conversely, let $K$ be a compact set that is not pluripolar. Since $\omega $ is Kähler, $0=V_\omega $ is a model potential for $\omega $ of mass $\int _ X\omega ^ n>0$, by proposition 16.21.2, and (16.27.6) for this model potential gives $\operatorname {Cap}_\omega (K)>0$. Since $\operatorname {Cap}_\omega (K)\leq G(\operatorname {Cap}_{\theta ,\psi }(K))$ and $G(0)=0$, it follows that $\operatorname {Cap}_{\theta ,\psi }(K)>0$. The assertion about Borel sets follows from the inner regularity in proposition 16.25.6.

(2) Let $\delta >0$ and $A_ j=\{ |f_ j-f|>\delta \} $. By the upper bound, $\operatorname {Cap}_\omega (A_ j)\leq c\sum _\alpha \operatorname {Cap}_{B'_\alpha }(A_ j\cap B_\alpha )$. Each term is at most the outer capacity $\operatorname {Cap}^*_{B'_\alpha }(\overline{B_\alpha }\cap A_ j)$, which tends to zero because $\overline{B_\alpha }$ is a compact subset of $B'_\alpha $. Hence $\operatorname {Cap}_\omega (A_ j)\to 0$, and therefore $\operatorname {Cap}_{\theta ,\psi }(A_ j)\to 0$.

(3) Let $f$ be quasi-continuous on $X$ and $\eta >0$. For each $\alpha $ there is an open set $W_\alpha \subset B'_\alpha $ with $\operatorname {Cap}_{B'_\alpha }(W_\alpha )<\eta $ such that $f$ is continuous on the compact set $\overline{B_\alpha }\setminus W_\alpha $. Put $W=\bigcup _\alpha W_\alpha $. Then $X\setminus W$ is the union of the finitely many closed sets $\overline{B_\alpha }\setminus W$, on each of which $f$ is continuous, so $f$ is continuous on $X\setminus W$; and $\operatorname {Cap}_\omega (W)\leq \sum _\alpha c\operatorname {Cap}_{B'_\alpha }(W_\alpha )<Nc\eta $, where $N$ is the number of balls. Now let $u\in \mathrm{PSH}(X,\theta ')$. By lemma 16.11.2(1), $u$ is quasi-continuous. Its pole set $P$ is pluripolar, because near each point $u$ is a plurisubharmonic function plus a smooth one (definition 6.14.3). Hence, by lemma 16.11.1(2) there are open sets $W'_\alpha \subset B'_\alpha $ with $P\cap B'_\alpha \subset W'_\alpha $ and $\operatorname {Cap}_{B'_\alpha }(W'_\alpha )<\eta $. Replacing $W_\alpha $, chosen as above for $f=u$, by $W_\alpha \cup W'_\alpha $, we obtain an open set $W$ with $\operatorname {Cap}_\omega (W)<2Nc\eta $ which contains $P$, such that $u$ is continuous on $X\setminus W$ and finite there. In both cases $\operatorname {Cap}_{\theta ,\psi }(W)\leq F(\operatorname {Cap}_\omega (W))\leq F(2Nc\eta )$, which is arbitrarily small when $\eta $ is small. $\square$