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18.8 A topology on ordinary singularity types8E117060E468

8E117060E468

The ordinary and the ideal envelope of a potential record different parts of its singularity. If a topology on ordinary singularity types requires both envelopes to converge, their equality becomes a closed condition, which every limit of analytic singularity types satisfies. For the converse, the dyadic approximants of a potential with equal envelopes must converge to it. We therefore add coordinates that these approximants control in a fixed Hermitian background: controlled inclusions of multiplier ideals, and the weighted total variation of divisorial Lelong numbers on each fixed modification. The closure of the analytic types is then exactly the set of types with equal envelopes, on the whole space of ordinary types.

This topology depends on the zero obstacle hidden in the envelopes, and a smooth change of background moves that obstacle. Testing all smooth obstacles removes this dependence, and raises the question whether equality of the envelopes below zero implies equality below every obstacle. Examples of zero volume built from divisors in fibrations and products over curves answer this question on explicit classes of potentials. On these classes, polar curvature on the curve factors also gives complete metrics, which we compare with the complete topologies known in positive volume at the end of the section. From 18.11 on, several constructions use the volume comparison of ordinary models and the curve models of section 13.16; those results do not depend on the present section, and the reader may consult them as they are cited.

Fix a Hermitian form $\omega $ on $X$ and write

\[ \mathfrak S(X,\omega )=\mathrm{PSH}(X,\omega )/{\sim }. \]

In this section $[u]$ denotes the ordinary class of $u$, that is, its class modulo bounded differences. For each smooth compact modification $\pi :Y\to X$, fix a Hermitian form $\gamma _\pi $ on $Y$. For a prime divisor $E\subset Y$, set $w_ E=\int _ E\gamma _\pi ^{n-1}$, and let $\nu _ E(u)=\nu (u\circ \pi ,E)$ be the generic Lelong number of section 5.38.

Lemma 18.8.1.8FA80F8B6F5A For $u,v\in \mathrm{PSH}(X,\omega )$, the sum

D93497D2DEF1 18.8.2
\begin{equation} d_\pi ([u],[v])=\sum _{E\subset Y\text{ prime}}w_ E|\nu _ E(u)-\nu _ E(v)| \end{equation}

is finite and defines a pseudometric on $\mathfrak S(X,\omega )$. Changing $\gamma _\pi $ gives an equivalent pseudometric for the fixed modification $\pi $.

Proof. The current $T=\pi ^*\omega +\mathrm{dd}^{\mathrm{c}}(u\circ \pi )$ is positive. Divisorial removal in lemma 5.38.1 gives, for every finite family of prime divisors,

\[ \sum _ Ew_ E\nu _ E(u)\leq \int _ YT\wedge \gamma _\pi ^{n-1}<\infty . \]

Taking the supremum over finite families shows that the sum converges and that only countably many coefficients are positive. The same bound holds for $v$. Bounded changes of $u$ or $v$ do not affect the coefficients, and the triangle inequality holds term by term. Finally, two Hermitian forms on the compact manifold $Y$ bound one another by fixed positive multiples, and so do the corresponding divisor weights. $\square$

For $t>0$ and $a>1$, let

91C58D64DE57 18.8.3
\begin{equation} \mathcal U(t,a)=\{ ([u],[v]):\mathcal I(atu)\subseteq \mathcal I(tv),\quad \mathcal I(atv)\subseteq \mathcal I(tu)\} . \end{equation}

These are relations between ordinary classes, since a bounded change leaves every multiplier ideal unchanged. Let $d_ I$ be the pullback of the metric (18.6.2), in background $\omega $, under $[u]\mapsto [u]_{\mathcal I}$, and put

\[ d_ O([u],[v])=\| P_\omega [u]-P_\omega [v]\| _{L^1(\mathrm dV)}. \]

Both are pseudometrics on ordinary classes: $d_ I$ compares the ideal envelopes, whereas $d_ O$ also detects differences of ordinary envelopes within a single multiplier-ideal class.

Proposition 18.8.4.B919F0D14EE2 Finite intersections of the relations $d_ O<r$, $d_ I<r$, $d_\pi <r$ and $\mathcal U(t,a)$, with $r,t>0$ and $a>1$, form a base of entourages for a uniformity on $\mathfrak S(X,\omega )$. Denote its topology by $\tau _{\omega ,0}$, where the subscript $0$ records the zero obstacle in the envelopes. A net $[u_ j]$ converges to $[u]$ if and only if both envelopes converge in $L^1$, $d_\pi ([u_ j],[u])\to 0$ for every fixed $\pi $, and $([u_ j],[u])\in \mathcal U(t,a)$ eventually for every fixed $t,a$. Smooth positive choices of $\mathrm dV$ and of the $\gamma _\pi $ do not change the uniformity.

Proof. All generators are symmetric and contain the diagonal. Multiplier ideals decrease as the weight increases, as one sees from local integrability after subtracting a local upper bound from the potential. For $b=\sqrt a$, put $\mathcal V=\mathcal U(t,b)\cap \mathcal U(bt,b)$. If $([x],[y]),([y],[z])\in \mathcal V$, then

\[ \mathcal I(atx)\subseteq \mathcal I(bty)\subseteq \mathcal I(tz), \]

and the same argument with $x$ and $z$ exchanged shows that $\mathcal V\circ \mathcal V\subseteq \mathcal U(t,a)$. For the pseudometric relations, halving the radius gives the corresponding refinement under composition. Taking finite intersections of these refinements proves the axioms of a uniformity, and convergence in it means eventual membership in each generating entourage. Bounded ratios of smooth positive volume forms and lemma 18.8.1 show that the auxiliary choices do not matter. $\square$

Each entourage involves only finitely many fixed modifications. The definition therefore requires neither a countable cofinal family of modifications nor convergence that is uniform over all modifications.

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