Theorem 11.41.3.BF6A67E9A40E Keep the line bundles and metrics fixed. Let nonzero sections $s_{jr}$ converge to $s_ r\ne 0$ in the finite-dimensional spaces of global sections, and let positive weights satisfy $t_{jr}\to t_ r>0$. Given Hermitian forms $\beta _ j$, define $\varphi _ j$ and $\theta _ j=\beta _ j+\sum _ rt_{jr}c_1(L_ r,h_ r)$ by the preceding formulas. Suppose either that $\beta _ j\to \beta $ in $C^\infty $ for a Hermitian form $\beta >0$ and $F_ j\to F$ in $C^\infty $, or that $\beta _ j=\beta $ for all $j$ and the tests $F_ j$ converge to $F$ in $L^1$ with a common $L^\infty $ bound. Then $\varphi _ j\to \varphi $ in $L^1$, and the solutions $u_ j\sim \varphi _ j$ for $(\theta _ j,F_ j)$, normalized by $\sup _ Xu_ j=0$, converge in $L^1$ to the similarly normalized solution $u\sim \varphi $. For the residual solutions $v_ j$ normalized to have mean zero, one also has
\[ \sup _ X(\varphi _ j+v_ j)\longrightarrow \sup _ X(\varphi +v). \]
The residuals converge smoothly in the first regime and uniformly in the second.
Proof.
The convergence of the residual solutions follows from proposition 11.40.4 in the first case and from the preceding compactness argument in the second. The fixed metrics and bounded weights give a common smooth lower bound for $\mathrm{dd}^{\mathrm{c}}\varphi _ j$. The convergent sections give a common upper bound for $\varphi _ j$ and convergence at a point outside all limiting divisors, so the suprema of the $\varphi _ j$ are bounded above and below. By theorem 5.25.2, every subsequence has a further subsequence converging in $L^1$. Off the limiting divisors, convergence of the sections and weights gives local uniform convergence to $\varphi $; since those divisors have zero volume, every $L^1$ cluster point equals $\varphi $. Thus $\varphi _ j\to \varphi $ in $L^1(X)$.
Put $C_ j=\sup _ X(\varphi _ j+v_ j)$ and $C=\sup _ X(\varphi +v)$. The sums converge in $L^1$ and have a common smooth lower bound for their complex Hessians, since $\mathrm{dd}^{\mathrm{c}}v_ j\geq -\beta _ j$ and the residual forms are uniformly bounded. Applying lemma 5.25.1 at maximum points gives $\limsup _ jC_ j\leq C$. A maximum point of $\varphi +v$ lies off the poles of $\varphi $, and evaluation there, using local uniform convergence, gives $\liminf _ jC_ j\geq C$.
It follows that $u_ j=\varphi _ j+v_ j-C_ j\to \varphi +v-C=u$ in $L^1$. For an empty family of divisors, the assertion reduces to the convergence of the residual solutions and of their suprema.
$\square$
Pome
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