24.8 Topologies on singularity types103C284C8243

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A topology on singularity types must keep information that weak convergence of currents can lose. Deficits of upper volume achieve this when bounded mass and positive lower volume provide uniform control, and on curves polar curvature gives a complete description even at zero volume. The problem is to extend such control to degenerations in higher dimension and to backgrounds with unbounded mass. Throughout this section, for a smooth real $(1,1)$-form $\theta $ and $u\in \mathrm{PSH}(X,\theta )$, $P_\theta [u]$ is the ordinary envelope of definition 5.40.4 and $P_\theta [u]_{\mathcal I}$ the multiplier-ideal envelope of definition 18.4.1. An ordinary model is a potential $u$ with $P_\theta [u]=u$, an ideal model is one with $P_\theta [u]_{\mathcal I}=u$, and an analytic model is an ordinary model with analytic singularities. The lower and upper singularity volumes $\underline{\operatorname {vol}}_\theta (u)\leq \overline{\operatorname {vol}}_\theta (u)$ are the infimum and the supremum of the non-pluripolar masses over the ordinary singularity type of $u$ (definition 16.28.58).

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.

24.8.2 Volume comparison and analytic approximationEB37AB0B2D63

In this subsection $X$ has bounded mass and $\theta $ is a smooth big background. For $u,v\in \mathrm{PSH}(X,\theta )$, the comparison $d_\theta (u,v)$ of definition 13.22.1 is the sum of the deficits of the upper volumes of $u$ and $v$ relative to the upper volume of $\max (u,v)$, and $\mathcal M_\delta (\theta )$ denotes the set of ordinary models with lower volume at least $\delta >0$. A common positive lower bound for the volumes converts a small deficit into a stretching estimate. The resulting quantitative $L^1$ bound, together with a modulus for three points that replaces the triangle inequality, gives separation and Cauchy completeness on $\mathcal M_\delta (\theta )$ (theorem 13.22.7); the cutoff must be closed, as example 13.22.8 shows. The positive lower bound is also what prevents the comparison from identifying distinct ordinary models: example 13.22.3 gives two distinct models at zero lower volume with comparison zero, in positive closed backgrounds in every dimension at least two and in positive nonclosed backgrounds in every dimension at least three. Rooftop continuity gives monotone lower bounds, which yield convergence in capacity (proposition 13.23.2, theorem 13.23.3).

Analytic approximation detects exactly the ideal models. Theorem 13.24.3 corrects analytic regularizations so that they stay in the fixed background; their ordinary models converge for $d_\theta $ to the ideal envelope, and their comparison with the original potential tends to the excess of the upper volume of its ideal envelope over its own upper volume. Conversely, at positive lower volume, theorem 13.24.5 shows that ideal models are closed under this convergence. Although $d_\theta $ fails the triangle inequality even on analytic models on a projective manifold, theorem 13.27.4 constructs a complete metric on the ordinary models of positive lower volume with the same convergent sequences; its Cauchy criterion also controls the reciprocals of the lower volumes, which excludes approach to zero volume. By corollary 13.27.9, the ideal models form a closed complete subspace, which is the completion of the analytic models of positive volume.

Monotone limits show where positivity enters. For decreasing ordinary models in backgrounds that decrease uniformly to a smooth big form, theorem 13.21.2 gives continuity of the lower volume even when the limiting volume is zero; if the limiting lower volume is positive, the upper volumes and the Lelong numbers on every fixed modification are continuous as well. For increasing sequences in a fixed background, theorem 13.21.1 gives continuity of the ordinary envelopes and of both volumes under the same positivity condition. For arbitrary decreasing potentials whose limit has positive lower volume, proposition 13.21.3 shows that the ordinary envelopes converge exactly when the upper volumes do. Likewise, when the ideal envelope has positive lower volume, proposition 13.24.2 shows that the ordinary and ideal envelopes of a potential agree exactly when the potential and its ideal envelope have the same upper volume.

There is an explicit divisorial positivity test on $\mathbb P^2$. Let $\kappa $ be the normalized Fubini–Study form, prescribe distinct hyperplanes $H_ j$ with summable coefficients $a_ j>0$, and put $A=\sum _ ja_ j$. If $\mathrm{dd}^{\mathrm{c}}\theta =0$ and $\theta -A\kappa >0$, equality with the profile $\sum _ ja_ j\operatorname {ord}_ E(\pi ^*H_ j)$ on every smooth compact modification specifies one ideal type. By 13.24.8, its ideal envelope is ordinarily equivalent to the logarithmic series, and every representative of this ordinary type has mass $\int _ X(\theta -A\kappa )^2$. The nonclosed example of 13.24.10 has infinitely many hyperplanes outside one pencil and positive lower volume. General positivity criteria from restricted ideals or divisorial orders, and the remaining zero-volume implications in 13.24.11, remain open.

The comparison is compatible with changes of background as long as these controls persist. A smooth gauge change, followed by ordinary projection, preserves it exactly (proposition 13.25.1). For a fixed potential and backgrounds decreasing to $\theta $, corollary 13.21.4 gives continuity of the lower volume and, at positive lower volume, of the ordinary envelope. The uniform estimate (13.25.4) allows the curvature errors of analytic regularization to tend to zero, as in theorem 13.25.5. For perturbations of the background in both directions, a convex correction produces admissible transported models, with a uniform comparison whose limiting notion of convergence does not depend on the chosen strict reference; this gives the analytic criterion of theorem 13.26.3, theorem 13.26.7. None of these arguments covers loss of bigness or the boundary at zero volume in higher dimension.

24.8.3 Polar curvatureB245A62A943C

Let $Y$ be a compact connected Riemann surface with a smooth positive form $\theta $. For $u\in \mathrm{PSH}(Y,\theta )$, the polar curvature $\mu _ u=\mathbf1_{\{ u=-\infty \} }(\theta +\mathrm{dd}^{\mathrm{c}}u)$ determines the ordinary model of $u$, and its atomic part determines the ideal model. By theorem 13.28.4, $d_\theta (u,v)$ is the total variation of $\mu _ u-\mu _ v$, also at zero non-pluripolar mass. By corollary 13.28.7, all normalized ordinary models form a complete geodesic space, and the ideal models form the closed subspace obtained by completing the analytic models of positive mass. On ideal models of positive mass this agrees with the topology of the source metric in theorem 20.3.21, corollary 20.3.22, for which this space is not totally bounded. Its completeness therefore has a different content from the compact completion obtained from weak convergence of traces.

Keeping only the atomic part of the polar curvature is a $1$-Lipschitz retraction onto the ideal models and selects the unique nearest ideal model; by proposition 13.28.11, the distance to the analytic closure is exactly the mass of the diffuse part of the polar curvature. The model of zero mass in example 13.28.14 has trivial multiplier ideals but positive distance from that closure. Convergence in total variation implies convergence in $L^1$ and in capacity, even at zero mass (theorem 13.28.8), whereas the moving pole of example 13.28.10 shows that convergence in capacity does not imply convergence in total variation.

Degeneration of the background is also explicit on curves. Between smooth backgrounds of positive area, theorem 13.29.3 gives correspondences of the model spaces that rescale by the ratio of the areas and are compatible with ideal projection, also at models of zero mass. Under uniform convergence of the backgrounds, the corresponding normalized representatives vary uniformly in $L^1$ and in capacity (corollary 13.29.7). When the limiting area is zero, theorem 5.29.23 shows that all normalized potentials converge to the unique limiting potential, uniformly over the admissible families, and the unscaled diameters of the model spaces tend to zero.

The joint space of theorem 13.30.3 treats the smooth background as a further coordinate. It is complete for the combination of smooth convergence of backgrounds and total variation of polar curvature, its analytic closure is exactly the ideal locus, even at zero area, and on each fixed background it restricts to the curve topology described above. In higher dimension, the realization of compatible singularity data and a replacement for the comparison of finite upper volumes are unresolved.

Finite curve maps give another compatibility within the curve theory. For $r:C'\to C$ of degree $d$, a smooth base form $\beta $ of nonnegative area and $\sigma =r^*\beta $, 18.24.1 commutes both envelopes with pullback below every pulled-back smooth obstacle. The polar metric scales by $d$, and 18.24.2 gives a $1$-Lipschitz retraction with polar measure $d^{-1}r^*r_*\mu $, preserving polar mass and commuting with ideal projection. By 18.24.3, this describes the closed complete subspace of models that descend to the original base of a compact curve map, with its original-base normalization. The cubic example shows that this averaging need not select a nearest model. These conclusions do not supply a complete topology for general higher-rank backgrounds.

In dimension at least two the polar part of the first curvature current loses information. For $u\in \mathrm{PSH}(X,\omega )$, write $\operatorname {Pol}_1(u)=\mathbf1_{\{ u=-\infty \} }(\omega +\mathrm{dd}^{\mathrm{c}}u)$, and $\operatorname {Pol}_\pi (u)$ for the corresponding current of $u\circ \pi $ on a modification $\pi $ of $X$. By proposition 18.26.2, distinct analytic ideal models with isolated logarithmic poles, whose multiplier ideals differ at suitable weights, all have $\operatorname {Pol}_1=0$. The blowup at the pole detects the coefficient on the exceptional divisor (corollary 18.26.4), but every fixed countable family of modifications misses some such models (corollary 18.26.5).

Tracing polar curvature against Gauduchon forms gives a uniform normalization over modifications. Fix a family $\mathscr R$ of smooth modifications $\pi \colon Y_\pi \to X$ and Gauduchon forms $\eta _\pi $ on $Y_\pi $. The comparison $\delta ^{\mathrm{pol}}_\omega $ of (18.26.7) is the supremum over $\pi \in \mathscr R$ of the total variation of the polar curvature of $u\circ \pi $ traced against $\eta _\pi ^{n-1}$, normalized by the trace of $\pi ^*\omega $. By proposition 18.26.8, it separates analytic models and, on curves, recovers the complete curve topology up to a constant factor. Bounded equivalence preserves polar currents (proposition 5.46.3), so transport by a smooth gauge change, followed by ordinary projection, preserves the comparison (corollary 18.26.10). Since polar curvature is closed for every pole set (proposition 5.46.7), equality of the scalar traces recovers the full polar currents on each modification (corollary 18.26.11). Arbitrary independent choices of the Gauduchon forms can nevertheless fail to induce the positive-volume topology, even for analytic ideal models on $\mathbb P^2$ (proposition 18.26.13). The space of polar trace coordinates is complete (proposition 18.26.9), and for sequences that are Cauchy for the comparison, proposition 18.26.12 gives compatible weak limits of the polar currents, all carried by the pole set of a single $\omega $-plurisubharmonic function. Whether these limiting currents are realized by an ordinary model is a further problem.

Even all modifications together do not make the first polar comparison separate models or give the right analytic closure. On every compact Kähler manifold of dimension at least two, proposition 18.26.17 constructs a nonzero ordinary model whose polar currents vanish on every modification, whose non-pluripolar volume is positive but strictly smaller than that of the zero model, and whose multiplier ideals are all trivial. More precisely, corollary 18.26.18 gives a complete ordered interval of such models with volumes $A-Mt^ n$ and mutual distances $M|s^ n-t^ n|$, where $A=\int _ X\omega ^ n$ and $M>0$ is a constant of the construction; for every $0<\delta <A$, the interval can be chosen inside $\mathcal M_\delta (\omega )$. Every first polar coordinate vanishes on this interval, and only its zero endpoint is ideal. Consequently every choice of Gauduchon forms identifies distinct models, and the closure of the analytic models for the polar comparison contains nonideal models.

The same diffuse cutoff can be placed in a Hermitian background. Proposition 18.26.19 separates the loss $Mt^ n$ in the mass of the potential from lower-degree terms coming from the background. If $\mathrm{dd}^{\mathrm{c}}(\omega ^ q)=0$ for $1\leq q<n$, the mass is the same for all bounded-equivalent representatives, and corollary 18.26.20 extends the whole interval of models to this Hermitian background; in particular the obstruction occurs on every compact complex surface with a Gauduchon form. For an arbitrary Hermitian background, proposition 18.26.21 computes the mass of each representative from finitely many torsion coefficients and the same loss $Mt^ n$. On a surface, corollary 18.26.22 shows that the mass is constant over the bounded-equivalence class exactly when $\omega $ is Gauduchon; otherwise even admissible smooth bounded perturbations change it, and the exact values would require optimizing these terms, which depend on the representative.

Under bounded mass and positive lower mass, proposition 18.26.23 still gives positive lower volume and a strict gap in upper volume for the diffuse ordinary models, while their first polar currents vanish on every modification. By the same proposition, this obstruction holds for every Hermitian metric on every compact complex surface. The theorem on preservation of residuals, theorem 13.19.6, distinguishes the diffuse parameters after ordinary projection. Consequently, in every background with both mass properties, corollary 18.26.24 gives a complete ordered interval of models throughout each range $0\leq t\leq t_0<1/C$, with $C$ the constant of proposition 18.26.19. Its upper-volume coordinate is strictly decreasing and Lipschitz, although the torsion terms prevent the explicit power formula from being inferred, and every first polar coordinate still vanishes. The same theorem preserves the entire residual measure, so its location matters: in proposition 18.26.25, two models carry residual mass $r$ in disjoint regions, have zero first polar data, and are at distance at least $2r$; when $\mathrm{dd}^{\mathrm{c}}(\omega ^ q)=0$ for $1\leq q<n$, their two singularity volumes also coincide. A complete extension must therefore distinguish this spatial residual information.

Even the full residual measure on the base is not enough. When $\mathrm{dd}^{\mathrm{c}}(\omega ^ q)=0$ for $1\leq q<n$, proposition 18.26.26 uses anisotropic maxima on one diffuse pole set to build a compact metric family of ordinary models indexed by positive weight vectors. Fixing the product of the weights gives complete subfamilies of real dimension $n-1$ whose members have the same residual measure, the same two singularity volumes, trivial multiplier ideals and zero first polar currents on every modification. The comparison distance nevertheless separates them, through the symmetric difference of the associated coordinate rectangles. A complete extension must therefore also detect singularity information lost by the full base residual. The proposition does not assert that the full residual products agree on every modification.

Return to a curve $Y$ with a smooth positive form $\theta $. On ideal models, the ordinary-type topology $\tau _{\theta ,0}$ is the topology of total variation of the atomic data, and its canonical metric includes the boundary of zero mass (theorem 13.28.17). On general ordinary models it is weaker than polar total variation: diffuse polar curvature can concentrate into an atom in the limit of the envelopes, which produces the Cauchy sequence without an ordinary-type limit in proposition 13.28.19. Characterizing the analytic closure on the whole ordinary space and extending the model topology to a complete one are therefore separate requirements.

Problem 24.8.4.35C633451730 For a smooth big background $\theta $ on a manifold with bounded mass, theorem 13.27.4 gives a complete metrizable topology on the ordinary models of positive lower volume, in which the closure of the analytic models is exactly the set of ideal models. Extend this topology to ordinary models beyond bounded mass and positive lower volume, keeping the closure of the analytic models exactly the set of ideal models. The extension should keep the compatibility with smooth gauge changes and with two-sided perturbations of the background established in section 13.25, section 13.26, and it should treat limits of backgrounds that lose bigness or a uniform positive lower bound for the volume. On curves it should recover the complete topology of polar total variation of corollary 13.28.7 on all models. Determine whether the extension in higher dimension is completely metrizable.

Any such extension must keep the singularity information that weak limits lose. Potentials of infinite mass are already covered by $\tau _{\omega ,0}$: for instance, the potential of proposition 18.5.5 has infinite mass, while its canonical envelopes, the ordinary and the ideal one, both equal zero. On curves, zero volume does not prevent completeness, but in higher dimension the stretching argument still needs a positive lower bound for the volume. There is nevertheless a restricted locus where a complete metric is known at zero volume. On the divisorial product locus of proposition 18.15.8, polar curvature on the curve factor gives a complete metric whose analytic closure is exactly the set of ideal models; the top-degree volume comparison vanishes identically on this locus and therefore cannot produce this metric. The same restricted metric and analytic closure exist in a positive nonclosed background on a projective manifold (corollary 18.15.9), and the locus, the metric and the ideal projection persist when the second curve coefficient vanishes and the limiting background is not big (corollary 18.15.10). Proposition 18.16.5 gives a complete sum metric, with the same properties of ideal projection and analytic closure, on the locus induced by several independent curve models, and corollary 18.16.6 preserves this locus and metric through a nonclosed auxiliary coefficient and its nonbig limit at zero. When the curve coefficients vary, proposition 18.17.3 gives compatible weighted metrics, discarding vanishing coordinates, with uniform $L^1$ collapse when all curve areas vanish, and proposition 18.17.5 compares these metrics quantitatively across different sets of active coordinates, with the exact distance for ordered coefficients. Proposition 18.18.3, Corollary 18.18.5 extend the construction to arbitrary smooth admissible curve backgrounds and identify the limiting singleton, and proposition 18.20.2 preserves the restricted complete metric through arbitrary compact complex auxiliary factors with semipositive vertical terms, independently of bounded mass on the ambient manifold.