25.11 Volumes under mass boundsD116D93B0EBE

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For a nonclosed background, the mass of a single potential does not determine the masses in its singularity type. The lower and upper volumes $\underline{\operatorname {vol}}_\theta (u)$ and $\overline{\operatorname {vol}}_\theta (u)$ of definition 16.28.58, the infimum and the supremum of the mass over the ordinary singularity type of $u$, record the information that a single mass loses. Under bounded mass, comparison of these volumes controls envelopes and defines a topology on models of positive lower volume. On curves, polar curvature describes the boundary where the lower volume vanishes.

25.11.1 Singularity volumes and relative classesFFA645E73814

The volumes used in section 13.17 are those of Boucksom, Guedj and Lu [Definition 3.1, BGL25], expressed in terms of potentials. For closed backgrounds the two volumes coincide, because the non-pluripolar mass is invariant under bounded changes of the potential, as recalled in [Example 3.2, BGL25]. In the big case, the monotonicity theorem theorem 13.17.1 is [Proposition 3.7, BGL25], and the argument for lower volumes in proposition 13.17.2 is Li's [Proposition 2.5, XL26].

The order test in 13.24.8 uses the projective valuative characterization of Darvas and Xia [Theorem 2.13 and Proposition 2.30, DX22] in a common closed Kähler background. Their Lelong convention differs from the divisor coefficient used here by a factor of two, which leaves equality of order profiles unchanged. The countable divisorial domination and envelope arguments are those of lemma 5.38.5, corollary 18.4.9; they identify the whole ordinary type before its mass is computed. The upper bound for the closed top mass uses Witt Nyström's monotonicity theorem [WN19], while the lower bound follows directly from local bounded truncations. Expansion with smooth corrections, as in [Section 2.1, BGL25], then gives the nonclosed mass throughout this type. This computation applies to the prescribed summable hyperplane profile with positive smooth residual; it does not assert mass invariance for arbitrary unbounded types in a pluriclosed background.

The relative full-mass class of definition 13.18.4 is [Definition 3.2, XL26]. The stretching criterion lemma 13.18.1 is [Lemma 3.1, XL26]; the proof given here keeps track of the deficit of upper volumes and chooses a representative whose mass is close to its upper volume. The case of zero deficit gives theorem 13.18.5, and a uniform bound for Lelong numbers turns the convex comparisons into the estimate of proposition 13.18.3.

The domination principle theorem 13.19.1 is [Proposition 3.2, XL26]; its proof uses that the reference has positive lower volume, but not that it is an envelope fixed point. The estimate for relative obstacles, lemma 13.19.4, is [Proposition 3.3, XL26]. Together they give the identity for upper volumes in [Lemma 3.2, XL26] and the identity for lower volumes in [Corollary 3.3, XL26], both stated in theorem 13.19.5. For $u\in \mathrm{PSH}(X,\theta )$, the proof first shows that the increasing limit of the obstacle envelopes has the ordinary singularity type of $P_\theta [u]$, so that domination can be applied before the equality of volumes is established. The resulting characterization of full mass gives the volume criterion in proposition 13.24.2.

When the pole set is compact of Hausdorff dimension zero and the lower volume is positive, theorem 13.19.6 shows that ordinary projection preserves the whole residual measure. Bounded mass gives uniform bounds for the traces of all local closed powers of the ordinary rooftop envelopes. Away from the pole set, convergence in capacity of bounded functions identifies the limits, and the Hausdorff estimate removes all mass of the lower-degree coefficients on the pole set; hence the lower-degree global pairings preserve the full total mass. Since the rooftop envelopes all have the same residual measure, positivity together with the local order of residuals forces the model to have this residual as well. The argument also identifies the weak limit of the full Monge–Ampère measures of these rooftop envelopes. The sources of the local inputs are given in section 25.3.

The construction of small stretches, lemma 13.20.1, is [Proposition 3.1, XL26], and its bound for mixed masses gives an explicit uniform scale of stretching. The additive estimate for lower volumes in proposition 13.20.6 is the case of smooth perturbations of [Proposition 3.3(ii), BGL25], with the mixed bound of [Lemma 2.4, BGL25] written in terms of potentials. The strict barrier gives the multiplicative comparison at positive lower volume, as in [Proposition 3.5(ii), Step 1, XL26]. Consequently, corollary 13.21.4 gives continuity of the lower volume of every fixed potential as the background varies, while continuity of the upper volume and of the ordinary envelope still requires positive lower volume.

The continuity along increasing sequences, theorem 13.21.1, is [Proposition 3.5(i), XL26]: contact identifies the limit of the envelopes and its upper volume, and the deficit estimate then shows that the approximating lower volumes are positive. The continuity along decreasing sequences of models, theorem 13.21.2, is [Proposition 3.5(ii), XL26], including its conclusion when the lower volume of the limit vanishes. The uniform small stretch shows that a positive lower bound along the models forces their limit to have positive lower volume, and strict corrections and tail envelopes reduce the variation of the background to the argument for a fixed background. Li states the result for perturbations $\theta +\varepsilon _ j\omega $ with $\varepsilon _ j\downarrow 0$, whereas the proof here treats arbitrary smooth backgrounds that decrease uniformly in the order of forms. The criterion for positive lower volume, proposition 13.21.3, is [Proposition 3.5(iii), XL26]. The deficit bound gives uniform convergence of Lelong numbers on each fixed compact model, and the envelopes of the restricted analytic approximants give the exact traces of corollary 13.21.5.

For semipositive backgrounds, proposition 13.20.4 is the forward direction of [Proposition 2.11, ALS26], deduced here from the strict barrier. This direction requires neither the identity for ordinary envelopes nor positivity of the lower mass $v_-(\theta )$ of definition 13.14.14; that assumption is used for the converse. Applied to an envelope fixed point, the proposition compares Li's condition (13.20.5) with the condition (16.28.67) of Alehyane, Lu and Salouf.

25.11.2 Model topology and analytic approximation572621232871

The comparison function $d_\theta $ of definition 13.22.1, defined through the upper volumes of two potentials and of their maximum, is introduced in [Section 4.2, XL26]. That it separates models follows from the characterization by envelopes. The normalized stretching argument of [Lemma 4.2, XL26] gives the quantitative $L^1$ bound in proposition 13.22.4, and the three-point estimate (13.22.6) passes convergence from a subsequence to the whole Cauchy sequence. The construction by maxima of tails in theorem 13.22.7 follows [Theorem 4.3, XL26], with a fixed $\delta >0$ and the non-strict lower bound $\underline{\operatorname {vol}}_\theta (u)\geq \delta $. The strict bound $>\delta $ used in version 1 of Li's preprint is not preserved under limits of Cauchy sequences, as example 13.22.8 shows. Li explicitly allows $d_\theta $ to violate the triangle inequality, and example 13.27.1 gives an example of such a failure with intersecting divisors.

The criterion for finite rooftop envelopes, lemma 13.23.1, is [Lemma 4.1, XL26]: after choosing representatives whose masses are close to their upper volumes, a collapse would contradict the lower volume of the common upper bound. Li's argument with softened stretches [Proposition 4.4, XL26] gives proposition 13.23.2; only the limiting potential needs positive lower volume, and the same comparison gives convergence of the lower volumes of the rooftop envelopes. The two stretching scales in [Theorem 4.4, XL26] produce the monotone bounds of theorem 13.23.3. Since finite rooftop envelopes are again normalized models, continuity along decreasing models passes their volume bounds to the infinite tails; monotone convergence in capacity then applies, and the subsequence criterion gives convergence of the original sequence.

The results on analytic approximation in section 13.24 combine this continuity of volumes with the limit of the dyadic multiplier-ideal envelopes. Mixing with a strict analytic potential, with rational coefficients, removes the curvature error in any smooth big background, and two comparisons of backgrounds then give the exact excess of upper volume in (13.24.4). Conversely, stretching and concavity of the multiplier-ideal envelope show that its fixed points are closed under the upper-volume comparison at positive lower volume. This proves the analytic criterion theorem 13.24.5 and the Cauchy completeness inherited there. The boundary where the lower volume vanishes in higher dimension, and positivity criteria in terms of ideals, are the separate questions of 13.24.11.

The comparisons of backgrounds in section 13.25 follow from the same arguments with volumes and envelopes. Transport by a smooth gauge preserves the competitor currents, and ordinary projection corrects the zero obstacle; the explicit scale of small stretches makes the strict barrier uniform on sets of models whose lower volumes are bounded below by a positive constant. This yields (13.25.4) and extends the analytic criterion to positive background errors that tend to zero. These statements concern singularity volumes and canonical models; they neither identify the mass of the unregularized envelope with a gauge invariant nor assert convergence of arbitrary representatives.

The two-sided transport in section 13.26 mixes the original model with a strict analytic potential. A common upper background bounds both this transport and the maximum of two transported models, and the resulting volume bounds give uniform comparison, convergence of the transported reference, and equivalence of the limiting comparisons obtained with different strict corrections. The estimate for nearby backgrounds and the three-point modulus then reduce the analytic criterion of that section to theorem 13.25.5.

The complete metric in section 13.27 is obtained by the standard construction with finite dyadic chains, carried out directly from the three-point modulus. Stretching first gives continuity of the lower volume, and the reciprocal of the lower volume controls the boundary where it vanishes; the statement about the analytic completion then follows from the approximation criterion above. This metric induces the topology of the upper-volume comparison; it does not turn the comparison function itself into a metric.

In dimension $n\geq 2$, polar curvature on all modifications still does not detect all ordinary models. The Cantor construction of example 13.28.14 gives the curve potentials of lemma 18.26.15; their maximum has vanishing first polar currents, but its truncations keep a positive top-degree mass $M$ near the common pole set. A rank bound for smooth maxima gives the construction in every dimension. By proposition 18.26.17, corollary 18.26.18, every compact Kähler manifold, with Kähler form $\omega $ and $A=\int _ X\omega ^ n$, carries a complete ordered interval of diffuse models $p_ t$ with volumes $A-Mt^ n$ and distances $M|s^ n-t^ n|$ between $p_ s$ and $p_ t$, and every choice of the first polar comparison collapses this interval. Hence the closure of the analytic models for the polar comparison contains models that are not ideal. The identities for invariant mass extend these formulas to the setting of corollary 18.26.20.

For an arbitrary Hermitian background with bounded mass and positive lower mass, the mass bounds for representatives give proposition 18.26.23. This applies to every Hermitian metric on a compact complex surface, by the existence of Gauduchon metrics, the independence of the bounded mass property from the metric (proposition 13.14.15) and corollary 12.19.15. Preservation of residuals and additivity of volumes then give corollary 18.26.24 for all parameters in the strict range. The upper volume along this interval is strictly decreasing and Lipschitz, although in general the length of the interval is not given by the power formula valid for closed backgrounds, and all first polar coordinates still vanish. In proposition 18.26.25, two cutoffs with disjoint supports have equal total residual masses and vanishing first polar data, yet their maximum is zero and their distance is positive; in backgrounds with invariant mass, both singularity volumes coincide as well. These scalar data therefore cannot recover the spatial distribution of the residual mass.

A direct calculation with maxima in lemma 18.26.16 identifies the full residual measure as the tensor product of the measures on the curve factors, multiplied by the product of the positive weights; for a smooth maximum the coefficient reduces to the volume of a simplex. In proposition 18.26.26, a common cutoff and a uniform curvature bound give a compact family of models indexed by a box of positive weights. Invariance under projection identifies the comparison distance in this family, up to a fixed scale, with the Lebesgue measure of the symmetric difference of coordinate rectangles. The subfamilies on which the product of the weights is constant are complete and have real dimension $n-1$, although their members have the same full residual measure on $X$, the same two singularity volumes, the same multiplier ideals and the same first polar currents. The construction takes place in a background with invariant mass and concerns the residual measure on $X$ itself; it does not identify the full residual products on every modification. The problem of describing the complete topology remains open.

25.11.3 Curves and zero-volume limits9AB6A1306853

The inverse maximum principle in the plane, lemma 5.27.14, is the two-dimensional case of the result stated by Brezis and Ponce [Theorem 3.1, BP04], who attribute it to Dupaigne and Ponce; the proof given here uses harmonic barriers on shrinking neighborhoods of a compact polar set. Applied to differences of potentials, it shows that the polar curvature measures are ordered as the potentials are, and subtracting a common Green potential gives the formula (5.27.16) for the maximum. Since these local facts are general, they are proved in section 5.21.

The classification of models on curves in section 13.28 combines this order of polar measures with the Green representation of subsection 20.3.9. Every prescribed polar measure gives an ordinary model equivalent to its Green potential, and the characterization of envelopes of positive mass, together with uniqueness of harmonic functions at zero mass, identifies every ordinary envelope with such a model. The formula for the maximum then turns the upper-volume comparison into the total variation of measures, and completeness, geodesics and the nearest ideal model follow from the corresponding operations on measures. The exact defect of analytic approximation is the nonatomic polar mass, which for the Cantor example is the full curvature mass.

For a finite map of curves, the fiber maximum in 5.43.15 follows from the local branched coordinates and the removable-singularity theorem for subharmonic functions. The weighted trace in 5.43.16 includes the local degrees at atoms; its agreement with current pullback follows from the Riesz representation on a disk and the Poincaré–Lelong formula. These elementary operations give the envelope identities of 18.24.1. Applying the curve classification gives the degree-scaled metric embedding and the averaging retraction of 18.24.2. The component factorization then gives the complete subspace of models descending to the original base in 18.24.3. The cubic example 18.24.4 distinguishes this averaging from the nearest ideal retraction.

The continuity statement for curves, theorem 13.28.8, uses the theorem on convergence in capacity at positive volume in the interior. At zero mass the polar curvature is the whole current, which gives convergence of the currents in total variation. The local estimate lemma 5.27.11, which follows from the Green representation on a disk, the mean-value estimate for harmonic functions and the weighted Chern–Levine–Nirenberg inequality, gives convergence in capacity at this boundary. Moving a pole distinguishes this topology from that of total variation, while the approximation by finitely many atoms in theorem 18.6.4 distinguishes weak convergence of the curvature currents from convergence in capacity.

The correspondences between backgrounds in section 13.29 follow from the same classification by polar measures, which smooth gauges extend to every smooth background of positive area on a curve. Changes of background with equal area preserve total variation, whereas rescaling the area gives homotheties compatible with extraction of atoms and hence with ideal projection. These correspondences are realized by a smooth gauge, obtained from the scalar Poisson equation, followed by ordinary projection. Comparing the new obstacle with its maximum and minimum gives a uniform bound, which yields continuity in $L^1$ and in capacity up to the boundary of zero mass. Rationality is needed only for the assertion that analytic singularities are preserved under rescaling of the area, because analytic singularities in this book have rational coefficients. At zero area, positivity and Stokes' theorem leave a single normalized potential. The general convergence statement theorem 5.29.23, proved in section 5.21, follows from compactness of normalized potentials, uniqueness of harmonic functions and the local estimate relating total variation and capacity. It also gives the collapse of the model representatives as the unscaled metric diameters tend to zero.