Theorem 13.28.171BAAC809D02C

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

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

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. $\square$