18.6.3 Completion on a curveEFFAC36297D5

Proof. We first show that the normalized Green potentials of finite atomic probability measures are ideal models, and then approximate the curvature measure of an arbitrary $u\in \mathcal K_\theta $ by such measures.

Let $\delta _ p$ denote the Dirac mass at $p$. For distinct points $p_1,\ldots ,p_ N\in Y$, consider a finitely supported probability measure

The Green-kernel construction gives a potential $q_\nu $ with

Indeed, choose a kernel with $\mathrm{dd}^{\mathrm{c}}_ xG(x,p)=\delta _ p-\theta $. The sum $\sum _ j a_ jG(x,p_ j)$ has curvature $\nu $, and subtracting its finite supremum gives $q_\nu $. In a coordinate $z$ centered at $p_ j$, the function $q_\nu $ is $a_ j\log |z|^2$ plus a smooth function, and it is smooth away from the finite support of $\nu $. Since all of its curvature lies on its pole set, its non-pluripolar measure is zero.

We claim that $P_\theta [q_\nu ]_{\mathcal I}=q_\nu $. Let $w\in \mathrm{PSH}(Y,\theta )$ satisfy $w\sim _{\mathcal I}q_\nu $. The Lelong-number criterion of theorem 5.37.3 gives $\nu (w,p_ j)=a_ j$, where $\nu (w,p_ j)$ is the Lelong number of $w$ at $p_ j$. On a curve, this Lelong number is exactly the mass of the atom at $p_ j$ of the positive measure $\theta +\mathrm{dd}^{\mathrm{c}}w$. By distributional Stokes, that measure has total mass one, so the prescribed atoms already exhaust its mass and $\theta +\mathrm{dd}^{\mathrm{c}}w=\nu $. Thus $\mathrm{dd}^{\mathrm{c}}(w-q_\nu )=0$, and the difference is a harmonic distribution, hence a smooth function that is constant on the compact connected curve. Among these translates, the greatest nonpositive one is $q_\nu $, which proves the claim. By proposition 18.4.4, every such $q_\nu $ is also an ordinary model.

Now let $u\in \mathcal K_\theta $. Its curvature $\nu =\theta +\mathrm{dd}^{\mathrm{c}}u$ is a positive probability measure. Choose finitely supported probability measures $\nu _ j$ converging weakly to $\nu $. For instance, partition the compact metric space $Y$ into finitely many Borel sets of diameter tending to zero, place the mass of each set at a chosen point of it, and omit the sets of mass zero; uniform continuity of each continuous test function gives weak convergence.

The weights can be made rational. For a partition with weights $a_1,\ldots ,a_ N$, choose a large integer $M$, replace $a_ i$ by $b_ i=\lfloor Ma_ i\rfloor /M$ for $i<N$, put $b_ N=1-\sum _{i<N}b_ i$, and discard zero weights. The total absolute change is at most $2(N-1)/M$, which can be made to tend to zero along the partitions, so weak convergence is preserved. Writing $b_ i=m_ i/M$, the corresponding Green potential has the analytic type

with smooth local remainders.

Let $q_ j=q_{\nu _ j}$. By normalized compactness, every subsequence has a further subsequence converging in $L^1$ to some $q\in \mathcal K_\theta $. Passing to the limit in the curvature identity in the sense of distributions gives

By the same harmonic argument, $q-u$ is constant, and the constant is zero because both suprema vanish. Every cluster value is therefore $u$, so the whole sequence converges to $u$ in $L^1$. This proves the density assertion and identifies the completion.

To see incompleteness concretely, choose a nonconstant smooth real function $f$. For sufficiently small $\varepsilon >0$, one has $\theta +\varepsilon \mathrm{dd}^{\mathrm{c}}f>0$, so $u=\varepsilon f-\sup _ Y(\varepsilon f)$ is a smooth, normalized and nonconstant element of $\mathcal K_\theta $. Its multiplier ideals are all trivial, so zero is a competitor and $P_\theta [u]_{\mathcal I}=0$. Thus $u\notin \mathfrak M_{\mathcal I}(Y,\theta )$, although $u$ lies in the completion. The set of ideal models is therefore not closed in its completion, and the metric is not complete. $\square$