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.