PomeProposition 13.28.19.4651CC70BDA5 For the Fubini–Study form $\theta $ of area one on $\mathbb P^1$, the space $(\mathfrak S(\mathbb P^1,\theta ),d^{\mathrm{ot}}_\theta )$ is incomplete. In fact, there is a $d^{\mathrm{ot}}_\theta $-Cauchy sequence without limit consisting of ordinary models of one fixed positive non-pluripolar mass, all with trivial multiplier ideals.
13.28.18 Concentration of diffuse polar curvature0F287C888CF7
The atomic total variation cannot be omitted from theorem 13.28.17. On $\mathbb P^1$ with its Fubini–Study form $\theta $ of area one, fix a rational $0<\tau <1$ and take the atomic probability measures $\nu _ j$ and the ideal models $r_ j=P_\theta [(1-\tau )q_ j]_{\mathcal I}$ of proposition 18.6.6, choosing the rational approximations so that the largest atom of $\nu _ j$ tends to zero. The ordinary and ideal envelopes of $r_ j$ both tend to zero in $L^1$, and for each $t>0$ and $a>1$, the pair $([r_ j],[0])$ eventually lies in $\mathcal U(t,a)$. The currents $\theta +\mathrm{dd}^{\mathrm{c}}r_ j$ converge weakly to $\theta $. Nevertheless,
\[ d^{\mathrm{ot}}_\theta ([r_ j],[0]) \geq \| D_{r_ j}\| _{\mathrm{TV}}=1-\tau >0. \]
Thus these weaker conditions do not imply convergence for $\tau _{\theta ,0}$, even at a fixed positive mass.
The opposite difficulty concerns diffuse polar mass: the pseudometric controls the ordinary envelope in $L^1$, but not the total variation of the diffuse measure itself.
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
\[ w_ j=\int _{\mathbb P^1}q_ y\, \mathrm d\sigma _ j(y). \]
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 $\mathfrak S(\mathbb P^1,\theta )$. $\square$
On ordinary models, $\tau _{\theta ,0}$ and the topology of the polar metric $d_\theta $ are also different. Choose the Cantor support of $\sigma $ inside a positive real interval in the affine coordinate $z$, away from zero. The rotations $R_ j(z)=\mathrm e^{\mathrm i\varepsilon _ j}z$, with $0<\varepsilon _ j\to 0$, move this support off itself. Put $p=p_{c\sigma }$ and $\widetilde p_ j=p\circ R_ j^{-1}$. By rotation invariance of the envelope, the $\widetilde p_ j$ are ordinary models with polar measures $c(R_ j)_*\sigma $, which have no atoms. Continuity of rotations in $L^1$ yields $d^{\mathrm{ot}}_\theta ([\widetilde p_ j],[p])\to 0$, whereas the polar measures have disjoint supports, so $d_\theta (\widetilde p_ j,p)=2c$ by theorem 13.28.4. Conversely, polar total variation controls both terms of (13.28.16), by theorem 13.28.8. Thus $\tau _{\theta ,0}$ is strictly weaker than the polar metric topology on ordinary models, although the two agree on ideal models. The failure of continuity under smooth gauge changes with a fixed zero obstacle (proposition 18.11.2) gives a further difference: the polar metric is invariant under such changes, by section 13.29.
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