Proposition 10.3.12984F6438B4B

Proof. Choose finitely many coordinate balls $B_ k$ covering $X$, smaller concentric balls $B'_ k$ still covering $X$, and smooth functions $h_ k$ near $\overline{B_ k}$ with $\mathrm{dd}^{\mathrm{c}}h_ k=\theta $ (lemma 6.14.2, applied on slightly larger concentric balls). For $\varphi \in \mathrm{PSH}(X,\theta )$ the functions $\varphi +h_ k$ are plurisubharmonic on $B_ k$.

Let $(\varphi _ j)$ be a sequence with $a\leq \sup _ X\varphi _ j\leq b$. The plurisubharmonic functions $\varphi _ j+h_ k$ are uniformly bounded above on each $B_ k$. Since $\varphi _ j$ is upper semicontinuous on the compact manifold $X$, it attains its supremum at a point $x_ j$, and after passing to a subsequence $x_ j\to x$, since $X$ is compact and metrizable. Choose $k$ with $x\in B'_ k$. On $B_ k$ the functions $\varphi _ j+h_ k$ cannot converge locally uniformly to $-\infty $, because their values at $x_ j$ are bounded below by $a+\inf _{B'_ k}h_ k$ for large $j$. By theorem 5.10.4, a further subsequence converges in $L^1_{\mathrm{loc}}(B_ k)$. If $B_ l$ meets $B_ k$, the functions $\varphi _ j+h_ l$ cannot tend to $-\infty $ locally uniformly on $B_ l$, since on the nonempty open set $B_ k\cap B_ l$ they differ by the smooth function $h_ l-h_ k$ from a sequence that converges in $L^1_{\mathrm{loc}}$; so again a subsequence converges in $L^1_{\mathrm{loc}}(B_ l)$. Since $X$ is connected, every ball is joined to $B_ k$ by a finite chain of overlapping balls, and after finitely many extractions the sequence converges in $L^1(B'_ l)$ for every $l$, hence in $L^1(X)$, to some $\varphi $. The limit satisfies $\theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq 0$ as a distributional limit of positive currents, and its canonical representative belongs to $\mathrm{PSH}(X,\theta )$.

Now let $\varphi _ j\to \varphi $ in $L^1(X)$, all in $\mathrm{PSH}(X,\theta )$; we show that $\sup _ X\varphi _ j\to \sup _ X\varphi $. The submean inequality for $\varphi _ j+h_ k$ on small balls shows that the $\varphi _ j$ are uniformly bounded above. Let $x_ j\to x$; we claim that $\varlimsup _ j\varphi _ j(x_ j)\leq \varphi (x)$. Indeed, choose $k$ with $x\in B_ k$, let $c>\varphi (x)$ be real, and use upper semicontinuity to choose a closed ball $K\subset B_ k$ around $x$ on which $\varphi <c$. The function $c+h_ k$ is continuous near $K$ and $\varphi +h_ k<c+h_ k$ on $K$, so the second part of theorem 5.10.4 gives $\varphi _ j+h_ k<c+h_ k$ on $K$ for large $j$; hence $\varphi _ j(x_ j)<c$ for large $j$.

Choose a subsequence along which $\sup _ X\varphi _ j$ tends to $\varlimsup _ j\sup _ X\varphi _ j$ and points $x_ j$ with $\varphi _ j(x_ j)=\sup _ X\varphi _ j$ converge to some $x$; the claim gives $\varlimsup _ j\sup _ X\varphi _ j\leq \varphi (x)\leq \sup _ X\varphi $. Conversely, choose a subsequence along which $\sup _ X\varphi _ j$ tends to $\varliminf _ j\sup _ X\varphi _ j$ and $\sum _ j\| \varphi _ j-\varphi \| _{L^1}<\infty $; by monotone convergence $\sum _ j|\varphi _ j-\varphi |$ is integrable, hence finite almost everywhere, so $\varphi _ j\to \varphi $ almost everywhere along it. We get $\varphi \leq \varliminf _ j\sup _ X\varphi _ j$ almost everywhere, hence everywhere, as noted at the beginning of this section. The two inequalities prove that the supremum is continuous; in particular, the limit in the previous paragraph satisfies $a\leq \sup _ X\varphi \leq b$. Thus the set in question is sequentially compact, hence compact, in the metric space $L^1(X)$. The continuous function $\| \cdot \| _{L^1}$ is bounded on the compact set of normalized functions.

The arguments so far use only that $X$ is a connected compact complex manifold of positive dimension and that the volume form defining $L^1(X)$ is smooth and positive; hence they also prove the last assertion of the proposition.

For smoothly convergent backgrounds, choose $C\geq 0$ with $\theta _ j\leq C\omega $ for all $j$. Then $C\omega +\mathrm{dd}^{\mathrm{c}}\varphi _ j\geq 0$, so all $\varphi _ j$ lie in $\mathrm{PSH}(X,C\omega )$, and the first part, applied to the closed form $C\omega $, gives an $L^1$-convergent subsequence and the convergence of the suprema. Since $\theta _ j+\mathrm{dd}^{\mathrm{c}}\varphi _ j\to \theta +\mathrm{dd}^{\mathrm{c}}\varphi $ as distributions, the limit belongs to $\mathrm{PSH}(X,\theta )$. $\square$