25.12.1 Equations of prescribed ordinary type7C4DEB2F0F6B

The equations of prescribed ordinary type in theorem 15.12.1 are [Theorems 5.4 and 5.7, ALS26]. Replacing their reference volume $\omega ^ n$ by $\mathrm{d}V$ multiplies the density by a smooth positive function and preserves its $L^ p$ class, and a change of the normalization of the complex Hessian by a positive constant is absorbed by rescaling both the potential and the exponent.

The two constructions of [Lemmas 5.5–5.6, ALS26] are the relative estimate theorem 15.12.3 and the rooftop supersolution lemma 15.12.4; together they keep the solution in the ordinary type of the reference. For a big background, the argument with bounded subsolutions of Guedj and Lu [Lemma 3.3 and Theorem 3.4, GL23E] gives lemma 15.3.1, theorem 15.3.3.

The trace calculation with a barrier in section 15.19 follows Guyett and Mc Gowan [v1, Proposition 3.2, pp. 7–12, GM26]. Their geometric setting is a small crepant resolution, in Fujiki's class $\mathcal C$, of a normal projective Calabi–Yau variety, and their abstract Theorem 3.1 treats a closed semipositive big background with an upper bound for the mass. In the book, the arguments with bounded subsolutions and exponential comparison of lemma 15.3.1, theorem 15.3.3 give uniform bounds for the potentials and the scalars when the density is a fixed smooth positive function. The trace argument keeps the torsion of the perturbation and also allows a fixed nonzero Chern–Ricci form. Its conclusion is convergence of the potentials along subsequences, in a semipositive background with an explicit strict analytic barrier. The assertions of that paper about metric completions and conifold contractions are not part of this argument.

The bundle and the contraction in 15.19.8 form the standard model of a node used in [v1, Equation (2.1), p. 4, and Section 4, pp. 15–16, GM26]. The logarithmic curvature identity (15.19.11) gives a strict barrier for this model directly, and a cutoff in a positive collar makes it global. This proves the convergence of the potentials and the assertion about the area of the exceptional curves for these local resolutions, without using a classification of small resolutions or a theorem on metric completions. Since the canonical bundle is torsion, its local holomorphic trivializations give the Chern–Ricci-flat density.

The equation on the singular space, (15.19.16), is obtained through the general correspondence of theorem 9.11.5 between potentials on a resolution and potentials on the singular space defined by disk inequalities. For closed backgrounds, the identification in [v1, Lemma 3.1 and the proof of Corollary 1.1, pp. 13–14, GM26] uses the equation on normal spaces and the estimate for adapted measures of Eyssidieux, Guedj and Zeriahi [arXiv v2, Theorem 6.3 and Lemma 6.4, pp. 28–29, EGZ09]. In the standard model of a node, (15.19.15) gives in lemma 15.19.14 the exact range $p<3$ of integrability and the weak $L^3$ endpoint, for every smooth positive density on the resolution. The bounded energy identity already used in theorem 9.11.43 proves theorem 15.19.17, using smooth strict positivity only away from the exceptional curves.

The arguments for the metric completion and the distance in lemma 15.19.21, proposition 15.19.22 are those of [v1, Lemma 5.2, pp. 23–24, and Proposition 5.1, pp. 24–26, GM26]. Compact collars in the regular part separate the points added in the completion, and the first and last visits of a path to the finitely many contracting regions give a distance error controlled by these regions; uniform convergence of the metrics on each compact subset of the regular part suffices. The hypotheses on the diameters of punctured neighborhoods in the limit and on the intrinsic diameters for the approximating metrics are stated explicitly; the conifold estimates and the asymptotics of cones needed to verify them are separate geometric inputs.

For curves, [v1, Lemma 4.1, p. 17, GM26] bounds the diameter of $\mathbb P^1$ by a multiple of $A^{1/3}$, where $A$ is the area, under a uniform upper bound for the metric. Averaging over straight coordinate segments and splitting the kernel given by the reciprocal of the distance give the sharper logarithmic estimate of lemma 15.19.24 on every fixed compact Riemann surface. The radial example in example 15.19.25 has the same order, and rescaling it shows that the area alone does not control the diameter. The two radial paths and the connection through the zero section in proposition 15.19.26 give the mechanism of [v1, Proposition 4.2, pp. 20–21, GM26], with the uniform bound on the restriction to the curve and the radial bounds stated as explicit hypotheses.

Strict singular barriers extend the comparison argument to potentials with minimal singularities. The development starting in section 15.13 follows [Section 4, BGL25], where the bounded mass assumption of the earlier sections of that paper is dropped. The equations with minimal singularities, culminating in theorem 15.16.7, are [Theorems 4.6–4.7, BGL25]. The solution has finite mass, although the masses in its ordinary singularity type are not assumed to be uniformly bounded.