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.