Proposition 11.45.18132C5E1825BF

Proof. Existence of the smooth positive weights is theorem 11.4.2, applied to the residual metrics $\Omega _ j$ on the surface. Put $\nu _ j=\rho _ j\Omega _ j^2$ and $L_ jh=2\Omega _ j\wedge \mathrm{dd}^{\mathrm{c}}h/\Omega _ j^2$. Distributional Stokes gives $\int L_ jh\, \mathrm{d}\nu _ j=0$. Uniform convergence to the positive form $\Omega $ supplies a common determinant lower bound and gives $L_ jh\to L_ uh$ uniformly for every fixed smooth $h$. Every weak cluster point of the probabilities $\nu _ j$ is therefore invariant for $L_ u$. By theorem 11.45.16, it equals $\nu $, proving convergence of the whole sequence. The same determinant convergence gives $F_ j\to F$ uniformly.

For a smooth real $(1,1)$ test form $\xi $, the ratios $\Omega _ j\wedge \xi /\Omega _ j^2$ converge uniformly to $\Omega \wedge \xi /\Omega ^2$. Consequently,

Similarly, the positive densities $p_ j=\Omega _ j^2/\mathrm{d}W$ converge uniformly to $p=\Omega ^2/\mathrm{d}W$. For every $g\in C(X)$,

These arguments require no pointwise control of the weights.

For existence, heat-smooth $A$ on the first sphere to positive smooth functions $A_ j$ of the same mean $a$, converging uniformly to $A$. Green inversion gives smooth mean-zero $w_ j(x)$ with $\mathrm{dd}^{\mathrm{c}}w_ j=(A_ j-a)\omega _1$ and $w_ j\to w_0$ uniformly. Their residuals converge uniformly to $\Omega $. Adding $\varepsilon _ j h(x,y)$ for a fixed smooth function depending on both variables and $\varepsilon _ j\to 0$ preserves positivity for all sufficiently small coefficients and supplies nonsymmetric examples. Each $u_ j$ retains the fixed logarithmic divisor, and its non-pluripolar equation has scalar $c_0$ and logarithmic density $F_ j$. $\square$