24.8.1 Weak convergence and singularity data163A8FDB4B95

Weak convergence alone keeps little singularity information. The complete Gauduchon trace metric of corollary 11.31.6 induces the $L^1$ topology on normalized potentials, and the ideal metric of proposition 18.6.1 is the $L^1$ distance between the canonical ideal representatives $P_\theta [u]_{\mathcal I}$. The completion of the ideal metric can acquire points that are not ideal envelopes of their own classes: on a curve, theorem 18.6.4 approximates every normalized potential by ideal envelopes with finitely many logarithmic poles and zero non-pluripolar mass. In particular, such envelopes can converge to a smooth potential with strictly positive curvature. This convergence does not hold in capacity, since damped continuity would then preserve their zero measures.

A positive lower bound on the mass does not repair this. In proposition 18.6.6, analytic ideal envelopes of one fixed positive mass converge in the ideal metric to the smooth model $0$, whose mass is larger. Nor can the normalization functional recover the lost data: on the closed model domain of theorem 11.42.1, it records exactly the mass. The examples with positive residual in example 11.42.3, example 11.42.4 have identical functionals and supporting sets but different poles, different multiplier ideals and different limits of normalized potentials, also for nonclosed backgrounds.

An order bound gives more control than weak convergence. Let $h$ be a Hermitian form, let $\epsilon _ j\geq 0$ tend to zero, and let $a_ j\in \mathrm{PSH}(X,\theta +\epsilon _ jh)$ satisfy the one-sided bound $u\preceq _{\mathcal I}a_ j$ in the ideal order of section 5.33, that is, $\mathcal I(ku)\subseteq \mathcal I(ka_ j)$ for all $k>0$. Then proposition 18.7.1 identifies the limit of the ideal envelopes from the convergence of divisorial Lelong numbers, and it shows that along such ordered analytic approximations, including dyadic regularization, the ordinary envelopes converge to $P_\theta [u]$ exactly when $P_\theta [u]=P_\theta [u]_{\mathcal I}$. Inclusions of multiplier ideals at fixed weights, as in corollary 18.7.2, give the required convergence of divisorial numbers. Without the order bound, $L^1$ convergence together with convergence of every divisorial number is not enough, by example 18.7.3. These tests make sense even when the upper volume is infinite. In contrast, in a Hermitian background analytic classes are dense for the ideal metric in the whole space (corollary 18.7.9), so density for that metric cannot characterize equality of the envelopes.

The finite damped measures of section 13.15 control nonlinear limits throughout the space of potentials, including potentials of infinite mass. Since their weak continuity already follows from convergence in capacity, requiring it in addition imposes no further condition on multiplier ideals.

For a Hermitian form $\omega $, the uniform topology $\tau _{\omega ,0}$ of section 18.8 on ordinary singularity types combines both envelopes with controlled inclusions of multiplier ideals and the total variation of divisorial orders on each fixed compact modification. By theorem 18.9.1, the closure and the sequential closure of the analytic classes are both the set of ordinary classes with $P_\omega [u]=P_\omega [u]_{\mathcal I}$. This characterizes the analytic closure in every Hermitian background, including classes represented by potentials of infinite mass. The topology is not $T_0$ on ordinary classes. Transport by a smooth gauge change of the background is exact when the obstacle changes with the background, but with the zero obstacle kept fixed it is discontinuous already on curves, even within a cutoff of positive mass (proposition 18.11.2).

The uniformity of 18.12, generated by the envelopes $P_{\omega ,F}[u]$ and $P_{\omega ,F}[u]_{\mathcal I}$ below all smooth obstacles $F$ (definition 5.40.4), is invariant under smooth changes of background, and its analytic closure consists of the classes for which the two envelopes agree below every smooth obstacle (theorem 18.12.3). On curves this is the same analytic locus, but by proposition 13.28.24 the topology is incomplete and strictly weaker than polar total variation. In higher dimension, membership of $u$ in the relative envelope class $\mathcal E_{\mathrm{env}}(X,\omega ,P_\omega [u]_{\mathcal I})$ suffices for equality below every obstacle (proposition 18.12.2), and under bounded mass and positive lower volume of the ideal envelope, equality at the zero obstacle is already sufficient (proposition 13.24.2).

The fixed stronger potential in theorem 18.13.1 gives an explicit family in the sufficient relative envelope class on every positive Hermitian background. Its poles may contain a prescribed countable dense set, while its multiplier ideals are trivial at every positive weight. By proposition 18.13.4, both envelopes equal the unrestricted envelope below every continuous obstacle. The stretching property is supplied by the construction rather than deduced from equality below zero.

Several geometric cases are known without the volume condition. For every admissible smooth background on a curve, corollary 13.29.2 shows that equality at the zero obstacle, equality at all smooth obstacles and membership in the relative envelope class of the ideal envelope are equivalent. On products with separated backgrounds, proposition 18.16.2 computes both envelopes coordinatewise below separated smooth obstacles, also for curve backgrounds of zero area, and proposition 18.19.2 identifies equality at the zero obstacle with equality at every smooth obstacle, including obstacles that mix the coordinates and factors of zero area. This equivalence persists after adding a nonclosed auxiliary coefficient and a unit divisor (corollary 18.19.3), and for arbitrary compact complex auxiliary factors with a semipositive vertical term, without any mass bound on the ambient manifold (corollary 18.20.3). For divisorial fibrations over curves, corollary 18.14.10 gives the same equivalence at zero model volume, even with poles along whole fibers or with disconnected fibers, by reducing the obstacles to the cover formed by the fiber components. In example 18.14.8 the ambient background fails bounded mass, while the base potential belongs to the relative envelope class required for equality at all obstacles. Whether equality at the zero obstacle implies equality at all obstacles at zero lower volume in general, including backgrounds with unbounded mass, is unresolved.