Corollary 5.39.8.D3B517FBCE4F Let $\omega _0$ be a Hermitian form on $X$ and $d_{\omega _0}$ the distance of the associated Riemannian metric. For each $j$, let $T_ j=\theta _ j+\mathrm{dd}^{\mathrm{c}}v_ j\geq 0$, where $\theta _ j$ is a smooth real $(1,1)$-form, $v_ j$ is continuous, and $T_ j$ is a smooth Hermitian form off a proper analytic subset $A_ j$. Assume
\[ |\theta _ j|_{\omega _0}\leq \Lambda ,\qquad |v_ j(x)-v_ j(y)|\leq m(d_{\omega _0}(x,y)), \]
with one fixed $\Lambda $ and one nondecreasing modulus $m$ tending to zero such that the function in (5.39.5) satisfies $D(r_0/2)<+\infty $. By theorem 5.39.6, the identity of $X\setminus A_ j$ extends to a continuous surjection of $X$ onto the metric completion of $X\setminus A_ j$; let $d_ j$ be the pseudometric on $X$ obtained by pulling back the completed distance. Then the diameters of the $d_ j$ are uniformly bounded, and a subsequence converges uniformly on $X\times X$ to a continuous pseudometric. If $T_ j\geq c\omega _0$ on the regular loci for one $c>0$, the limit is a metric and dominates $\sqrt c\, d_{\omega _0}$.
Proof.
A finite fixed chart cover and the theorem give common constants $C,r_1>0$ such that
\[ d_ j(x,y)\leq CD(Cd_{\omega _0}(x,y)), \qquad d_{\omega _0}(x,y)<r_1. \]
For a pair meeting $A_ j$, approximate it by regular pairs in the manifold topology. The completed distance and $D$ are continuous, so the same estimate holds for all such pairs in $X\times X$. No uniform bound on the geometry of the analytic sets enters the constants. Dividing a minimizing geodesic of $d_{\omega _0}$ into a uniformly bounded number of short pieces and using the triangle inequality gives a uniform diameter bound. The triangle inequality also gives
\[ |d_ j(x,y)-d_ j(x',y')| \leq d_ j(x,x')+d_ j(y,y'), \]
so the distance functions are equicontinuous. The Arzelà–Ascoli theorem gives a subsequence $d_{j_ k}$ converging uniformly on $X\times X$ to a function $d_\infty $. Nonnegativity, symmetry, vanishing on the diagonal and the triangle inequality pass to the limit. The lower bound $T_ j\geq c\omega _0$ gives $d_ j\geq \sqrt c\, d_{\omega _0}$, first for regular pairs and then for all pairs, which proves the last assertion.
Without this lower bound the limit may degenerate. For $\varepsilon _ j\downarrow 0$, take $\theta _ j=\varepsilon _ j\omega _0$ and $v_ j=0$, with $A_ j=\varnothing $. These data satisfy the hypotheses with $m(r)=r^2$, and $d_ j=\sqrt{\varepsilon _ j}\, d_{\omega _0}$ converges uniformly to zero.
$\square$
Pome
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