Proposition 13.28.194651CC70BDA5

Proof. Choose a nonatomic polar probability measure $\sigma $ with compact support in an affine chart, using the Cantor construction in example 13.28.14. Fix $0<c<1$ and a point $x$ in that chart. Translating and shrinking the support towards $x$ gives nonatomic polar probability measures $\sigma _ j$ supported in disks about $x$ with radii tending to zero. Let $p_ j=p_{c\sigma _ j}$ be the ordinary models of lemma 13.28.1. Their polar measures are $c\sigma _ j$, their masses are $1-c$, and $D_{p_ j}=0$. Hence their Lelong numbers vanish, their multiplier ideals are trivial, and their ideal envelopes are zero.

We claim that $p_ j\to q_{c\delta _ x}$ in $L^1$. Write $q_ y=q_{c\delta _ y}$. Rotations of $\mathbb P^1$ preserving $\theta $ and sending $x$ to $y$ can be chosen to depend continuously on $y$ near $x$. They transport $q_ x$ to $q_ y$, so continuity of translations in $L^1$ gives $q_ y\to q_ x$ in $L^1$ as $y\to x$. Set

The common $L^1$ bound for normalized potentials makes this integral well defined. The function $w_ j$ is nonpositive, its curvature current dominates $c\sigma _ j$, and $w_ j\to q_ x$ in $L^1$. By (13.28.3), $p_ j$ is the upper envelope of all $v\leq 0$ with $\theta +\mathrm{dd}^{\mathrm{c}}v\geq c\sigma _ j$, so $w_ j\leq p_ j$.

By normalized compactness, every subsequence of $p_ j$ has a further subsequence converging in $L^1$ to some $v\leq 0$. The inequalities $\theta +\mathrm{dd}^{\mathrm{c}}p_ j\geq c\sigma _ j$ pass to the limit and give $\theta +\mathrm{dd}^{\mathrm{c}}v\geq c\delta _ x$. Since $q_ x=p_{c\delta _ x}$, the same envelope description gives $v\leq q_ x$. The lower bound $w_ j\leq p_ j$ gives $v\geq q_ x$ almost everywhere, hence everywhere. This identifies every cluster point and proves the claim.

Since $P_\theta [p_ j]=p_ j$ and $D_{p_ j}=0$, the classes $[p_ j]$ form a $d^{\mathrm{ot}}_\theta $-Cauchy sequence. A limit $[v]$ would satisfy $D_ v=0$ and $P_\theta [v]=q_ x$. This is impossible: the ordinary envelope preserves all multiplier ideals and hence all atoms, whereas $q_ x$ has an atom of mass $c$ at $x$. The sequence therefore has no limit in $\mathfrak S(\mathbb P^1,\theta )$. $\square$