Theorem 10.9.19E2AF1B6D37EC

Proof. The compact manifold $X$ has finitely many connected components, which are open, and it suffices to construct $\Phi $ on each of them, since the intersection of $A$ with a component is pluripolar in that component. On a component of dimension zero, which is a point, pluripolar sets are empty and $\Phi =0$ works. We may therefore assume that $X$ is connected of dimension $n\geq 1$, and we fix a Hermitian form $\omega $ on $X$ (one exists by section 5.6), so that the notation of this section applies.

Every point of $X$ has a coordinate ball $U$ and a plurisubharmonic function on $U$ whose pole set contains $A\cap U$, by definition 5.47.2; choose a smaller ball $U_1\Subset U$ around the point. By compactness finitely many of the balls $U_1$ cover $X$. By the last assertion of lemma 10.9.13 it therefore suffices to prove the following: if $U$ is a coordinate ball, $\varphi $ is plurisubharmonic on $U$ and $U_1\Subset U$ is a smaller ball, then $E=\{ \varphi =-\infty \} \cap U_1$ is globally pluripolar.

If $E$ is empty, there is nothing to prove, so let $E\neq \varnothing $. Choose balls $U_1\Subset U_2\Subset U_3\Subset U$ and subtract a constant from $\varphi $ so that $\varphi \leq 0$ on $U_3$. Suppose, to reach a contradiction, that $E$ is not globally pluripolar. By lemma 10.9.13, there is a constant $C$ with $0\leq T_ E^*\leq C$. From the extremal functions of compact sublevel sets of regularizations of $\varphi $, which have small capacity, we shall obtain a bounded function in $\mathrm{PSH}(X,\omega )$ with zero Monge–Ampère mass.

Step 1. Sublevel sets and their extremal functions. Convolution with radial mollifiers of radii decreasing to zero and smaller than the distance from $U_2$ to $\partial U_3$ gives smooth plurisubharmonic functions $\varphi _ j$ on $U_2$ that decrease to $\varphi $ as $j\to \infty $, by proposition 5.9.2(2). Since $\varphi \leq 0$ on $U_3$, we have $\varphi \leq \varphi _ j\leq 0$ on $U_2$, so $\| \varphi _ j\| _{L^1(U_2)}\leq \| \varphi \| _{L^1(U_2)}<\infty $. For integers $N,j\geq 1$ put

Then $K_{j,N}$ is compact and contains the open set $O_{j,N}$. Fix $x_0\in E$. Since $\varphi _ j(x_0)\downarrow -\infty $, there is $j_0(N)$ such that $x_0\in O_{j,N}$ for $j\geq j_0(N)$; we choose $j_0(N)$ nondecreasing in $N$. For these $j$ the set $K_{j,N}$ has nonempty interior, so $u_{j,N}=T_{K_{j,N}}^*$ is a bounded nonnegative function in $\mathrm{PSH}(X,\omega )$ by lemma 10.9.13. Every competitor for $T_{K_{j,N}}$ is nonpositive on the open set $O_{j,N}$; hence so is $T_{K_{j,N}}$, and its upper semicontinuous regularization satisfies $u_{j,N}=0$ on $O_{j,N}$.

For fixed $N$, the sets $K_{j,N}$ increase with $j$, because $\varphi _ j$ decreases. A competitor for $T_{K_{j+1,N}}$ is a competitor for $T_{K_{j,N}}$, so $u_{j,N}$ decreases with $j\geq j_0(N)$; let $u_ N$ be its limit. By lemma 10.9.3(3), $u_ N\in \mathrm{PSH}(X,\omega )$, and $0\leq u_ N\leq u_{j_0(N),N}$ is bounded. If $x\in U_1$ and $\varphi (x)<-N$, then $x\in O_{j,N}$ for all large $j$, so $u_ N(x)=0$. In particular $u_ N=0$ on $E$, so $u_ N$ is a competitor for $T_ E$, and

For fixed $j$, the sets $K_{j,N}$ decrease as $N$ increases, so $u_{j,N}\leq u_{j,N+1}$ for $j\geq j_0(N+1)$; letting $j\to \infty $ gives $u_ N\leq u_{N+1}$. Let $u_\infty $ be the upper semicontinuous regularization of $\lim _ Nu_ N$. By lemma 10.9.3(1), $u_\infty \in \mathrm{PSH}(X,\omega )$ and $0\leq u_\infty \leq C$.

Step 2. Convergence of the masses. By lemma 10.9.3(3), applied to the uniformly bounded decreasing sequence $(u_{j,N})_{j\geq j_0(N)}$ and to the uniformly bounded increasing sequence $(u_ N)_{N\geq 1}$, and testing the weak convergence with the constant function $1$ on the compact manifold $X$, we obtain

Moreover $\int _ X\omega _{u_\infty }^ n>0$ by proposition 10.9.10, since $u_\infty $ is bounded.

Step 3. The capacity of $K_{j,N}$. Choose a smooth function $h$ on $U_2$ with $\mathrm{dd}^{\mathrm{c}}h\geq \omega $ and $\sup _{U_2}|h|<\infty $, for instance a large multiple of $|z|^2$. If $v\in \mathrm{PSH}(X,\omega )$ and $0\leq v\leq 1$, then $v+h$ is plurisubharmonic on $U_2$ with $|v+h|\leq 1+\sup _{U_2}|h|$, and $\omega _ v^ n\leq (\mathrm{dd}^{\mathrm{c}}(v+h))^ n$ on $U_2$ by lemma 10.9.3(2). Since $K_{j,N}\subset \overline{U_1}$ and $-\varphi _ j\geq N$ on $K_{j,N}$, while $-\varphi _ j\geq 0$ on $U_2$, the weighted Chern–Levine–Nirenberg inequality (theorem 7.4.1), applied in $U_2$ to $\varphi _ j$, the compact set $\overline{U_1}$ and a compact neighborhood $L\subset U_2$ of $\overline{U_1}$, gives

The constant $C'$ depends on $\| \varphi _ j\| _{L^1(L)}\leq \| \varphi \| _{L^1(U_2)}$ and on the bound for $|v+h|$, but not on $v$, $j$ or $N$. Taking the supremum over $v$, we obtain $c_\omega (K_{j,N})\leq C'/N$.

Step 4. Conclusion. By proposition 10.9.16,

For fixed $N$, the uniformly bounded functions $u_{j,N}$ decrease to $u_ N$, so $u_{j,N}\to u_ N$ in $L^1(X)$ by dominated convergence, and $\sup _ Xu_{j,N}\to \sup _ Xu_ N$ by lemma 10.9.3(4). Letting $j\to \infty $ and using $u_ N\leq C$, we obtain $\int _ X\omega _{u_ N}^ n\leq \max \{ 1,C\} ^ nC'/N$. Letting $N\to \infty $ gives $\int _ X\omega _{u_\infty }^ n=0$, which contradicts Step 2. Hence $E$ is globally pluripolar, and the theorem is proved. $\square$