5.47.1 Pluripolar sets and relative extremal functions27FCEC6F336F

Proof. The boundary behavior of $H_ K$ follows from (5.47.3). Since $H_ K$ is bounded, its Monge–Ampère measure is defined (definition 7.2.1).

We first show that $(\mathrm{dd}^{\mathrm{c}}H_ K)^ n$ vanishes on $B\setminus K$. Choose competitors $v_ j$ for $K$ as in (5.47.4), and let $D$ be a ball with $\overline D\subset B\setminus K$. Let $\widehat v_ j$ be the function obtained from $v_ j$ by corollary 7.7.5: it is plurisubharmonic on $B$, equals $v_ j$ outside $D$, and satisfies $\widehat v_ j\ge v_ j$ and $(\mathrm{dd}^{\mathrm{c}}\widehat v_ j)^ n=0$ on $D$. Each $\widehat v_ j$ is again a competitor for $K$. Indeed, $\widehat v_ j=v_ j\le -1$ on $K$. Moreover $\widehat v_ j\le 0$: outside $D$ because $\widehat v_ j=v_ j$, and on $D$ by the maximum principle (proposition 3.4.7), since by upper semicontinuity the upper limit of $\widehat v_ j$ at each point $x\in \partial D$ is at most $\widehat v_ j(x)=v_ j(x)\le 0$.

Since $\widehat v_ j$ is increasing in $v_ j$ by corollary 7.7.5, the functions $\widehat v_ j$ increase with $j$, and $v_ j\le \widehat v_ j\le h_ K\le H_ K$ shows that $\widehat v_ j\to H_ K$ almost everywhere. They take values in $[-1,0]$, so theorem 7.6.1 gives $(\mathrm{dd}^{\mathrm{c}}\widehat v_ j)^ n\to (\mathrm{dd}^{\mathrm{c}}H_ K)^ n$ weakly, and therefore $(\mathrm{dd}^{\mathrm{c}}H_ K)^ n=0$ on $D$. Covering the open set $B\setminus K$ by countably many such balls $D$, we conclude that $(\mathrm{dd}^{\mathrm{c}}H_ K)^ n$ is carried by $K$.

As $h_ K=-1$ on $K$, the Borel set $K\cap \{ H_ K>-1\} $ is contained in $\{ h_ K<H_ K\} $, which has outer capacity zero by (5.47.5). Because $H_ K+1$ is a competitor for $\operatorname {Cap}_ B$, this set is therefore null for $(\mathrm{dd}^{\mathrm{c}}(H_ K+1))^ n=(\mathrm{dd}^{\mathrm{c}}H_ K)^ n$. Thus $(\mathrm{dd}^{\mathrm{c}}H_ K)^ n$ is carried by $K\cap \{ H_ K=-1\} $.

We now prove the capacity formula. As $H_ K+1$ is a competitor for $\operatorname {Cap}_ B$ and $(\mathrm{dd}^{\mathrm{c}}H_ K)^ n$ is carried by $K$, we have $\int _ B(\mathrm{dd}^{\mathrm{c}}H_ K)^ n=\int _ K(\mathrm{dd}^{\mathrm{c}}(H_ K+1))^ n\le \operatorname {Cap}_ B(K)$. Conversely, let $v$ be a competitor for $\operatorname {Cap}_ B$, let $0<\varepsilon <1/2$, and put $w=(1-2\varepsilon )(v-1)-\varepsilon $. Then $-1+\varepsilon \le w\le -\varepsilon $. In particular $w>-1$, so $K\cap \{ H_ K\ge w\} \subset K\cap \{ H_ K>-1\} $; this set has outer capacity zero and is therefore null for $(\mathrm{dd}^{\mathrm{c}}w)^ n=(1-2\varepsilon )^ n(\mathrm{dd}^{\mathrm{c}}v)^ n$. Moreover $H_ K(z)\to 0$ as $z\to \partial B$ while $w\le -\varepsilon $, so $\varliminf _{z\to x}(H_ K(z)-w(z))\ge \varepsilon >0$ for every $x\in \partial B$, and the comparison principle (theorem 7.3.11) applies to $H_ K$ and $w$ on $B$. We obtain

Letting $\varepsilon \downarrow 0$ and taking the supremum over $v$, we get $\operatorname {Cap}_ B(K)\le \int _ B(\mathrm{dd}^{\mathrm{c}}H_ K)^ n$.

Now let $(K_ j)$ be compact subsets of $B$ decreasing to $K$. Since $K\subset K_{j+1}\subset K_ j$, the functions $H_{K_ j}$ increase and are bounded above by $H_ K$. Put $H=(\sup _ jH_{K_ j})^*$. By theorem 5.10.2, $H$ is plurisubharmonic and equals $\sup _ jH_{K_ j}$ almost everywhere, and $H\le H_ K$ because $H_ K$ is upper semicontinuous. For the reverse inequality, let $v$ be a competitor for $K$ and let $0<\varepsilon <1$. The open set $G=\{ v<-1+\varepsilon \} $ contains $K$. The compact sets $K_ j\setminus G$ decrease and have empty intersection, so one of them is empty; hence $K_ j\subset G$ for all large $j$. For such $j$ the function $v/(1-\varepsilon )$ is a competitor for $K_ j$, and hence $v\le (1-\varepsilon )H_{K_ j}\le (1-\varepsilon )H$. Letting $\varepsilon \downarrow 0$ gives $v\le H$, and taking the supremum over all competitors $v$ for $K$ gives $h_ K\le H$. As $H$ is upper semicontinuous, $H_ K\le H$. Therefore $H=H_ K$, and $H_{K_ j}\to H_ K$ almost everywhere.

By theorem 7.6.1, $(\mathrm{dd}^{\mathrm{c}}H_{K_ j})^ n\to (\mathrm{dd}^{\mathrm{c}}H_ K)^ n$ weakly. All these measures are carried by the fixed compact set $K_1$, so testing with a function in $C_ c(B)$ equal to one on $K_1$ shows that their total masses converge. By the capacity formula, $\operatorname {Cap}_ B(K_ j)\to \operatorname {Cap}_ B(K)$, and this convergence is decreasing because capacity is increasing.

Finally, $\operatorname {Cap}_ B(K)\le \operatorname {Cap}_ B^*(K)$ because capacity is increasing. For the converse we may assume $K\neq \varnothing $, since both capacities of the empty set vanish. Choose $\delta >0$ such that the compact sets $K_ j=\{ z:\operatorname {dist}(z,K)\le \delta /j\} $ lie in $B$. They decrease to $K$, and each contains $K$ in its interior. Hence $\operatorname {Cap}_ B^*(K)\le \operatorname {Cap}_ B(\operatorname{Int}K_ j)\le \operatorname {Cap}_ B(K_ j)$, and the right-hand side tends to $\operatorname {Cap}_ B(K)$ by what we have just proved. $\square$

Proof. Choose compact sets $K_ j$ with $K_ j\subset \operatorname{Int}K_{j+1}$ and $\bigcup _ jK_ j=G$. Since the $K_ j$ increase, the functions $H_{K_ j}$ decrease. They take values in $[-1,0]$, so their limit $H$ is plurisubharmonic by theorem 5.10.1. Since $h_{K_ j}=-1$ on $K_ j$ and the upper regularization at a point only involves the values of $h_{K_ j}$ near that point, we have $H_{K_ j}=-1$ on $\operatorname{Int}K_ j$. Every point of $G$ lies in $\operatorname{Int}K_ j$ for all large $j$, so $H=-1$ on $G$. Thus $H$ is a competitor for $G$, and $H\le h_ G$. Conversely, every competitor for $G$ is a competitor for each $K_ j$, hence bounded above by every $H_{K_ j}$ and thus by $H$; so $h_ G\le H$. We have shown $h_ G=H$. As $H$ is upper semicontinuous, $H_ G=h_ G^*=H$; in particular $H_ G=-1$ on $G$. The boundary behavior follows from (5.47.3).

By theorem 7.3.7, $(\mathrm{dd}^{\mathrm{c}}H_{K_ j})^ n\to (\mathrm{dd}^{\mathrm{c}}H_ G)^ n$ weakly. By proposition 5.47.6, these measures are carried by $K_ j\subset \overline G$. Hence the limit measure vanishes on the open set $B\setminus \overline G$, that is, it is carried by the compact set $\overline G$, and testing with a function in $C_ c(B)$ equal to one on $\overline G$ shows that the total masses converge. Since the Borel sets $K_ j$ increase to $G$, proposition 7.4.3(2) gives $\operatorname {Cap}_ B(K_ j)\to \operatorname {Cap}_ B(G)$. Together with the capacity formula of proposition 5.47.6 for the $K_ j$, this yields $\operatorname {Cap}_ B(G)=\lim _ j\int _ B(\mathrm{dd}^{\mathrm{c}}H_{K_ j})^ n=\int _ B(\mathrm{dd}^{\mathrm{c}}H_ G)^ n$. $\square$

Proof. We prove (2)$\Rightarrow $(1), then (1)$\Rightarrow $(3) for relatively compact sets $E$ and (3)$\Rightarrow $(2), deduce (1)$\Rightarrow $(2) in general by exhaustion, and finally prove (2)$\Rightarrow $(3). The remaining assertions then follow by localization.

(2)$\Rightarrow $(1). We show more generally that every pluripolar set $E\subset B$ satisfies (1); a set as in (2) is pluripolar. By definition 5.47.2, every point of $B$ has an open neighborhood $W\subset B$ carrying a plurisubharmonic function whose pole set contains $E\cap W$. The second assertion of proposition 7.4.4, applied with $D=B$, therefore gives $\operatorname {Cap}_ B^*(E\cap L)=0$ for every compact set $L\subset B$. Exhausting $B$ by countably many compact sets and using countable subadditivity, we obtain $\operatorname {Cap}_ B^*(E)=0$.

(1)$\Rightarrow $(3) when $E\Subset B$. Fix an open set $O$ with $\overline E\subset O\Subset B$. Since $\operatorname {Cap}_ B^*(E)=0$, there are open sets $G_ j'\supset E$ with $\operatorname {Cap}_ B(G_ j')<1/j$. The open sets $G_ j=O\cap G_1'\cap \cdots \cap G_ j'$ decrease, contain $E$, lie in $O$, and satisfy $\operatorname {Cap}_ B(G_ j)\to 0$. Since the $G_ j$ decrease, the functions $H_{G_ j}$ increase; put $H=(\sup _ jH_{G_ j})^*$. By theorem 5.10.2, $H$ is plurisubharmonic, takes values in $[-1,0]$, and is the almost-everywhere limit of the $H_{G_ j}$. By proposition 5.47.7, $\int _ B(\mathrm{dd}^{\mathrm{c}}H_{G_ j})^ n=\operatorname {Cap}_ B(G_ j)\to 0$, and by theorem 7.6.1 these measures converge weakly to $(\mathrm{dd}^{\mathrm{c}}H)^ n$. Hence $\int \chi \, (\mathrm{dd}^{\mathrm{c}}H)^ n=\lim _ j\int \chi \, (\mathrm{dd}^{\mathrm{c}}H_{G_ j})^ n\le \lim _ j\operatorname {Cap}_ B(G_ j)=0$ for every $\chi \in C_ c(B)$ with $0\le \chi \le 1$, and therefore $(\mathrm{dd}^{\mathrm{c}}H)^ n=0$.

Since every $G_ j$ lies in $O$, the barrier (5.47.3), with a constant $A$ chosen for $O$, bounds every $H_{G_ j}$, and hence $H$, from below by one function tending to zero at $\partial B$; thus $H(z)\to 0$ as $z\to \partial B$. The last assertion of the comparison principle (theorem 7.3.11), applied to $u=H$ and $v=0$, gives $H\ge 0$, and therefore $H=0$. Finally, $E\subset G_ j$ gives $H_{G_ j}\le H_ E$ for all $j$; since $H_ E$ is upper semicontinuous, $0=H\le H_ E\le 0$.

(3)$\Rightarrow $(2). Suppose $H_ E=0$, and choose competitors $v_ j$ for $E$ as in (5.47.4). Fix a point $a\in B$ with $v_ j(a)\to 0$, and choose indices $j_1<j_2<\cdots $ with $v_{j_ l}(a)>-2^{-l}$. Since every term is nonpositive, the partial sums of $V=\sum _ lv_{j_ l}$ form a decreasing sequence of nonpositive plurisubharmonic functions, and $V(a)\ge -1$. By theorem 5.10.1, $V$ is plurisubharmonic on $B$; moreover $V\le 0$, and $V=-\infty $ on $E$ because every term is at most $-1$ there.

(1)$\Rightarrow $(2) in general. Let $\operatorname {Cap}_ B^*(E)=0$, and let $(L_ l)$ be an exhaustion of $B$ by compact sets. Each set $E\cap L_ l$ has outer capacity zero and is relatively compact in $B$, so the two preceding steps give $V_ l\in \mathrm{PSH}(B)$ with $V_ l\le 0$ and $E\cap L_ l\subset \{ V_ l=-\infty \} $. The pole sets of the $V_ l$ are Lebesgue null, so there is a point $a\in B$ at which all $V_ l$ are finite. Choose constants $c_ l>0$ with $c_ l|V_ l(a)|\le 2^{-l}$. The partial sums of $\sum _ lc_ lV_ l$ decrease and are at least $-1$ at $a$, so, as in the previous step, $V=\sum _ lc_ lV_ l$ is plurisubharmonic by theorem 5.10.1, and it is nonpositive. Every point of $E$ lies in some $L_ l$, where $c_ lV_ l=-\infty $ while all other terms are nonpositive; hence $V=-\infty $ on $E$.

(2)$\Rightarrow $(3). If $V$ is as in (2), then $\varepsilon V$ is a competitor for $E$ for every $\varepsilon >0$, so $\varepsilon V\le h_ E\le 0$. Letting $\varepsilon \downarrow 0$, we get $h_ E=0$ on the set $\{ V>-\infty \} $. This set has full measure and is therefore dense in $B$, so every neighborhood of a point contains points where $h_ E=0$. Hence $H_ E=h_ E^*\ge 0$, and $H_ E=0$. This completes the proof of the equivalences.

By the proof of (2)$\Rightarrow $(1), every pluripolar subset of $B$ satisfies (1), and hence all three conditions; in particular, it is globally pluripolar in $B$. Now let $X$ be a complex manifold and $E=\bigcup _ mE_ m$, where each $E_ m\subset X$ is pluripolar. Let $x\in X$, and let $B$ be a coordinate ball around $x$, identified with an open ball in $\mathbb C^ n$. Each $E_ m\cap B$ is pluripolar in $B$, so $\operatorname {Cap}_ B^*(E_ m\cap B)=0$ for every $m$, and $\operatorname {Cap}_ B^*(E\cap B)=0$ by countable subadditivity. By (1)$\Rightarrow $(2), the set $E\cap B$ lies in the pole set of a plurisubharmonic function on the connected coordinate neighborhood $B$ of $x$. Hence $E$ is pluripolar.

Finally, let $W\subset X$ be open and let $E\subset W$ be pluripolar in $W$. That $E$ is pluripolar in $X$ is not immediate from definition 5.47.2 at the boundary points of $W$, where a single plurisubharmonic function on a neighborhood must have poles at all nearby points of $E$. Let $x\in X$, and let $U$ be a coordinate ball around $x$, identified with an open ball in $\mathbb C^ n$. The open set $W\cap U$ is a countable union of balls $D_ k$ with $\overline{D_ k}\subset W\cap U$. Each set $E\cap D_ k$ is pluripolar in $U$: at points of $\overline{D_ k}\subset W$ we may use, after restriction to small balls in $U$, the functions provided by the pluripolarity of $E$ in $W$, and every point of $U\setminus \overline{D_ k}$ has a neighborhood disjoint from $D_ k$. By the previous paragraph, applied in $U$, the countable union $E\cap U=\bigcup _ k(E\cap D_ k)$ is pluripolar in $U$, and hence, by the first part of the theorem, it lies in the pole set of a plurisubharmonic function on $U$. Thus $E$ is pluripolar in $X$. $\square$