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.