5.25 Normalized compactness0B216F0B911E

Proof. On a coordinate ball about $x$, add a common smooth correction $q$ so that $u_ j+q$ and $u+q$ are plurisubharmonic. For a fixed small radius, the submean inequality bounds $(u_ j+q)(x_ j)$ by the average of $u_ j+q$ over the ball of that radius centered at $x_ j$. As $j\to \infty $, these averages converge to the average of $u+q$ over the ball of the same radius centered at $x$: the contribution of $u_ j-u$ tends to zero in $L^1$, and the integral of the fixed locally integrable function $u+q$ over a ball depends continuously on its center. Letting the radius decrease to zero proves the inequality.

The same estimate gives local uniform upper bounds for the $u_ j$ with $j$ large. By coefficient freezing and the local upper-envelope theorem, the tail envelopes are therefore quasi-plurisubharmonic, with the same lower Hessian bound. They decrease with $j$. If their limit exceeded $u(x)$ at some point $x$, the definition of the regularization would give indices $k_ j\geq j$ and points $x_ j\to x$ contradicting the first assertion for the subsequence $(u_{k_ j})$. On the other hand, an almost everywhere convergent subsequence of $(u_ j)$ shows that $h_ j\geq u$ almost everywhere, hence everywhere by lemma 5.24.3. Thus $h_ j\downarrow u$. $\square$

Proof. Let $(u_ j)$ be a sequence in $\mathcal K_\theta $. By upper semicontinuity, there are points $x_ j$ with $u_ j(x_ j)=0$; passing to a subsequence, we may assume that $x_ j\to x$. Cover $X$ by finitely many connected coordinate balls, and on a slightly larger ball around each of them choose a fixed smooth function $r_ i$ such that every $u_ j+r_ i$ is plurisubharmonic; such corrections exist because $\theta $ is smooth. Since $u_ j\leq 0$, these local functions are uniformly bounded above.

On a ball containing $x$ in its interior, locally uniform convergence to $-\infty $ is impossible, because the values $(u_ j+r_ i)(x_ j)=r_ i(x_ j)$ stay bounded below. Local compactness therefore gives a subsequence converging in $L^1$ on this ball. If a second ball meets the first, local compactness on the second ball cannot yield convergence to $-\infty $ on compact subsets of the intersection, since the first limit is finite almost everywhere there. Thus a further subsequence converges in $L^1$ on the second ball. Since $X$ is connected, the intersection graph of the finite cover by connected balls is connected. Proceeding along this graph and passing to subsequences finitely many times, we obtain local $L^1$ convergence on every ball, and a partition of unity then gives convergence in $L^1(X)$ to a function $u$.

The inequalities $\theta +\mathrm{dd}^{\mathrm{c}}u_ j\geq 0$ pass to the limit in the sense of distributions. By proposition 5.22.2, $u$ has a $\theta $-plurisubharmonic representative, still denoted by $u$. Since $u_ j\leq 0$, we have $u\leq 0$ almost everywhere, hence everywhere by lemma 5.24.3.

By lemma 5.25.1, $0=\lim _ j u_ j(x_ j)\leq u(x)$. Thus $u(x)=0$, and the limit belongs to $\mathcal K_\theta $. This proves sequential compactness, which is compactness in the metric space $L^1(X)$. The norm bound follows, since a compact subset of $L^1(X)$ is bounded. $\square$

Proof. On each connected coordinate ball, after a common smooth correction, the limit of a decreasing sequence of plurisubharmonic functions is either plurisubharmonic or identically $-\infty $. Fatou's lemma, applied to the negative parts, excludes the second alternative because of the $L^1$ bound. Hence $u$ is quasi-plurisubharmonic, and the inequality $\theta +\mathrm{dd}^{\mathrm{c}}u\geq 0$ passes to the decreasing limit. Finally, the nonnegative functions $u_1-u_ j$ increase to $u_1-u$, so monotone convergence gives $u_ j\to u$ in $L^1$. $\square$