25.12 Equations with prescribed singularitiesEFAA097B03EC

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Relative envelopes allow the solution of an equation to keep prescribed singularities while the measure on the right-hand side changes. Equations with $L^ p$ data in a prescribed ordinary type, equations with general measures, and equations under mass bounds on the metric use this framework in different ways. They differ both in the estimates available and in the singularity type of the solution obtained.

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.

25.12.2 General measures in semipositive backgrounds4D81B44E0A9B

The local decomposition of measures used in lemma 5.27.10 is Cegrell's [Theorem 6.3, Ceg98], on a bounded hyperconvex domain in dimension at least two; for a non-pluripolar measure its pluripolar part vanishes. The global constructions in lemma 15.24.4, proposition 15.24.1 patch these local representations together.

The bounded-subsolution theorem theorem 15.10.1, taken from [Theorem 5.1, ALS26], solves the exponential equation whose right-hand side has a bounded density with respect to the Monge–Ampère measure of a bounded potential. Truncating the density in such a representation of a general measure brings this theorem into the construction for general measures, and domination then allows passage to the limit. The domination theorem for finite measures, theorem 15.37.6 in section 15.37, is [Theorem 3.14, ALS26]: the fixed finite measure prevents collapse, whereas the other members of the envelope class need not have finite mass. The proof of theorem 15.37.10 also allows the measure being compared to be infinite. The singular strict domination principle, theorem 15.2.2, follows [Theorem 3.11, ALS26].

The homogeneous range result theorem 15.25.1 is the sufficient direction of Kołodziej and Nguyen's [Theorem 1.3, p. 2952; proof pp. 2958–2963, KNHolder18]. Their weighted regularization estimate is [Lemma 2.8, p. 2957, KNHolder18]. The proof here first approximates a Borel density by continuous functions and then chooses smooth references diagonally. Common Hölder bounds give uniform capacity estimates and a weighted modulus for the varying solution measures; bounds above and below for the normalization constants prevent degeneration before the uniform limit is taken.

The radial bounded product in lemma 5.27.26 is a direct Bedford–Taylor calculation: the logarithmic derivative of a smooth radial maximum fixes the spherical mass. Summing these local references with geometric weights and controlling a common cutoff gives proposition 15.26.2. The homogeneous range theorem yields corollary 15.26.3, with a normalized Hölder solution and an unknown positive coefficient. The bounded-reference theorem yields theorem 15.26.4 for bounded densities against the fixed spherical probability, including singular data on backgrounds with zero lower mass. This is an individual geometric sufficient condition, distinct from solvability for every non-pluripolar measure and from the exact density-threshold question.

Passing to the limit in the measures requires that the lower envelopes do not collapse. The continuity theorem for measures dominated by a fixed measure, theorem 15.37.19, is [Lemma 4.2 and Theorem 4.1, condition (2), ALS26]; in this case positive lower mass prevents collapse, and no upper bound on the mass is used. The argument with a common envelope preceding corollary 15.37.24 gives instead the first alternative of [Lemma 4.2, ALS26]. When the approximants have the ordinary singularity type of the reference, this alternative needs no assumption of positive lower volume.

A common barrier in the ordinary singularity type also gives corollary 15.9.4, the corresponding case of [Theorem 4.1, ALS26]. The reference need not be an envelope fixed point; the condition in a smaller background suffices. Integrals of bounded potentials against varying mixed measures are handled by the torsion calculation in lemma 5.44.7, which follows Kołodziej and Nguyen [Lemma 2.3, KN22].

The resulting equations with general finite non-pluripolar measures, theorem 15.37.4, are [Theorem 5.8 and Corollary 5.9, ALS26]. The constructions with a minimal and with a prescribed reference in subsection 15.37.27, subsection 15.37.30 use the approximations of the density from that proof, and they assume both the condition (16.28.67) in a smaller background and positive lower mass. The solutions have finite mass, but for arbitrary measures they need not differ boundedly from the reference. For a Hermitian background and the minimal reference, corollary 15.37.32 identifies the scalar in the equation with the density threshold of theorem 15.28.4, that is, the largest coefficient for which the scaled density admits a subsolution. The question that remains, 24.2.2 in section 24.1, concerns zero lower mass. The weighted addition theorem theorem 15.11.1 is [Theorem 5.2, ALS26]; the proof given here localizes the passage to the equation.

25.12.3 Big backgrounds under mass boundsBCD8D94DC861

Li's theory of equations with general measures [Theorems 6.1–6.2 and Corollary 6.1, XL26] assumes that the manifold has bounded mass and that the Hermitian metric has positive lower mass. It gives theorem 10.31.3 and the constructions with a model reference in subsection 10.31.43. Under bounded mass, positivity of the lower volume of the model reference implies that the metric has positive lower mass, so expressing the hypothesis through the reference reorganizes this setting without enlarging it.

The relative foundations of this theory use the domination, stretching and obstacle estimates discussed in section 25.11. In (10.31.27), the required rooftop envelope is constructed before lemma 13.19.4 is applied to the relative projection.

Moment and continuity estimates connect these operations on envelopes with existence. The compactness argument with first moments, using (10.31.10), follows [Proposition 4.2, XL26]: one chooses a reference potential in the same singularity type whose mass is sufficiently close to the upper volume, which need not be attained. Continuity for a fixed dominating measure is [Theorem 4.2, XL26]; there the finite rooftop envelopes use only the envelope class, while the two-sided bounds for the mass prevent collapse and control the tails.

The closure theorem for varying references, theorem 10.31.30, is [Corollary 4.1, XL26]. Projection to ordered lower models gives a fixed relative class for each tail of the sequence, and an exponential contact estimate prevents collapse. The stability for positive exponents in [Corollary 6.2, XL26] assumes uniformly bounded $L^ p$ data and the ordinary type of the reference. Theorem 10.31.47 treats instead the relative full-mass class: the models converge for the volume comparison with a common positive lower bound for their lower volumes, and the densities converge in $L^1$ with respect to a fixed finite non-pluripolar measure. Extracting a summable subsequence produces the dominating measure needed for the closure theorem. For a positive exponent, uniqueness gives convergence in capacity of the whole sequence of potentials. For exponent zero, the scalars and the measures converge, every $L^1$ cluster point of the normalized potentials solves the limiting equation, and the corresponding subsequence converges in capacity. In both cases, identifying the right-hand sides in $L^1$ gives convergence of the measures in total variation.

For the minimal reference, the equations with general measures in subsection 10.31.21 are [Theorems 5.2–5.3, XL26]. The estimate for higher moments in subsection 10.31.13 follows the concave-weight argument of [Proposition 5.1 and Lemma 5.1, XL26], with the integrability against the fixed singular measure stated explicitly. The smoothing of measures used in the existence proof is collected in lemma 5.44.12, lemma 5.44.10.

The conclusion of corollary 10.31.56, that for $L^ p$ data the solutions have the ordinary type of the reference, is [Corollary 6.1, XL26]. The relative estimate proposition 10.31.54 starts from a solution in the envelope class and uses a uniform moment of order $m>2n+1$. The concave-weight method is related to [Theorem 5.1, XL26], which already assumes the ordinary type of the reference and uses the moment exponent $m>n$; that theorem is not used to deduce the ordinary type of an arbitrary solution in the envelope class.

Addition of measures requires a common lower bound during truncation. The weighted construction in subsection 10.31.37 is [Proposition 6.1, XL26]. For unrestricted addition, (10.31.50) follows [Theorem 6.3, XL26]; an additional rooftop envelope gives the common lower bound needed to pass through the cutoff.