We now compare the polar metric with the topology on ordinary singularity types introduced in section 18.8. That topology also uses the ordinary envelope, but it measures the total variation only of the atomic part of the curvature. On ideal models the atomic part is the whole polar measure, so the two topologies agree there. On arbitrary ordinary models, diffuse polar mass can concentrate without being detected by the atomic total variation.
We keep $Y$, $\theta $, $A$ and the decomposition $\mu _ u=D_ u+\sigma _ u$, and we use $\eta =\theta /A$ for $L^1$ norms. Let $\mathfrak S(Y,\theta )=\mathrm{PSH}(Y,\theta )/{\sim }$ be the space of ordinary singularity types, and write $[u]$ for the class of $u$. Recall from proposition 18.8.4 that $\tau _{\theta ,0}$ is the uniform topology on $\mathfrak S(Y,\theta )$ generated by the $L^1$ distances between ordinary envelopes and between ideal envelopes, by the total variation of divisorial Lelong numbers on each fixed modification, and by the controlled ideal relations $\mathcal U(t,a)$ of (18.8.3). For ordinary classes put
D1D81F67E5DB
13.28.16
\begin{equation} d^{\mathrm{ot}}_\theta ([u],[v]) =\| P_\theta [u]-P_\theta [v]\| _{L^1(\eta )} +\| D_ u-D_ v\| _{\mathrm{TV}}. \end{equation}
Theorem 13.28.17.1BAAC809D02C The finite pseudometric $d^{\mathrm{ot}}_\theta $ induces $\tau _{\theta ,0}$ on $\mathfrak S(Y,\theta )$. For $0\leq s\leq A$, the classes of ideal models
\[ \mathcal C_ s=\{ [q_ D]:D\text{ positive atomic},\ D(Y)\leq s\} \]
form a complete metric space both for $d^{\mathrm{ot}}_\theta $ and for the distance $\| D-E\| _{\mathrm{TV}}$ between $[q_ D]$ and $[q_ E]$, and these two metrics induce the same topology. The space $\mathcal C_ s$ is not totally bounded if $s>0$, and $\mathcal C_0=\{ [0]\} $.
Proof.
A smooth compact modification of a smooth curve is an isomorphism, and the divisor weight of a point is one. Hence the divisorial distance in proposition 18.8.4 is exactly the atomic total variation, and the two summands of (13.28.16) are among the coordinates defining $\tau _{\theta ,0}$.
Conversely, suppose that a net $[u_ j]$ converges to $[u]$ for $d^{\mathrm{ot}}_\theta $. The ideal envelope of $u$ is $q_{D_ u}$, by lemma 20.3.20. Convergence of the atomic parts in total variation implies $L^1$ convergence of these envelopes, by theorem 13.28.4, theorem 13.28.8. These results also apply to nets: the map $D\mapsto q_ D$ goes between metric spaces, so sequential continuity is equivalent to continuity. It remains to check the controlled ideal relations.
At a point $x$ with local coordinate $z$, the multiplier ideals on the curve are
\[ \mathcal I(tu)_ x=(z^{\lfloor tD_ u(\{ x\} )\rfloor }). \]
This follows from theorem 5.37.3, lemma 5.37.1: on a smooth curve the Lelong number at $x$ determines the ideal, and the logarithmic germ with that Lelong number has the displayed ideal. Write $D_ j=D_{u_ j}$ and $D=D_ u$, and fix $t>0$ and $a>1$. The set $F=\{ x:D(\{ x\} )\geq 1/(2t)\} $ is finite. At its points, $aD_ j(\{ x\} )\geq D(\{ x\} )$ and $aD(\{ x\} )\geq D_ j(\{ x\} )$ eventually. Outside $F$, the bound
\[ \sup _ x|D_ j(\{ x\} )-D(\{ x\} )|\leq \| D_ j-D\| _{\mathrm{TV}}\longrightarrow 0 \]
makes both $\mathcal I(tu)_ x$ and $\mathcal I(tu_ j)_ x$ unit ideals for large $j$. Hence both inclusions in (18.8.3) hold eventually at every point. This proves that $d^{\mathrm{ot}}_\theta $ induces $\tau _{\theta ,0}$.
The map $D\mapsto q_ D$ is injective, so $d^{\mathrm{ot}}_\theta $ is a metric on $\mathcal C_ s$. The positive atomic measures of mass at most $s$ are closed in total variation: a limit is positive, satisfies the mass bound, and is carried by the countable union of the sets of atoms of the sequence. Completeness of the space of finite signed measures therefore gives completeness for total variation. A $d^{\mathrm{ot}}_\theta $-Cauchy sequence in $\mathcal C_ s$ is Cauchy in total variation; its limit is again atomic, and theorem 13.28.8 gives convergence of the envelope coordinate. This proves completeness for $d^{\mathrm{ot}}_\theta $ as well as the equality of the two topologies.
If $s>0$, fix $0<b\leq s$ and infinitely many distinct points $x_ j$. The measures $b\delta _{x_ j}$ have pairwise total-variation distance $2b$, so $\mathcal C_ s$ is not totally bounded for either metric. For $s=0$ only the zero measure remains.
$\square$
By lemma 20.3.20, for $0\leq \delta \leq A$ the set $\mathcal C_{A-\delta }$ consists exactly of the classes of ideal models of mass at least $\delta $. The whole set $\mathcal C_ A$, including the models of zero mass, is complete. Its subset of positive mass is incomplete: for a point $x$, the measures $D_ j=A(1-1/j)\delta _ x$ form a Cauchy sequence whose limit $A\delta _ x$ has zero mass.
For general ordinary classes, the corresponding statement concerns a pseudometric. Put
\[ \mathcal G_{\theta ,\delta } =\left\{ [u]:P_\theta [u]=P_\theta [u]_{\mathcal I},\quad \int _ Y\mathrm{MA}_\theta (P_\theta [u]_{\mathcal I})\geq \delta \right\} . \]
For $0\leq \delta \leq A$ this set is complete for $d^{\mathrm{ot}}_\theta $, since every class $[u]$ in it is at distance zero from $[q_{D_ u}]\in \mathcal C_{A-\delta }$; limits need not be unique as ordinary classes. The set is not totally bounded when $\delta <A$. At $\delta =A$ the pseudometric vanishes identically on it, although it contains distinct classes by proposition 18.10.1. The union of these sets over all $\delta >0$ is incomplete, by the same one-atom sequence.
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.