25.1.16 Envelopes and model potentials3F57C7AFD15C

Envelopes with obstacles, rooftop envelopes and envelopes of singularity types are used throughout [XiaSampleBook]. The argument for lower semicontinuous obstacles and the two reductions for quasi-continuous obstacles are in [Lemma 2.4.3 and Theorem 2.4.4, XiaSampleBook]. We follow them in lemma 16.13.3, theorem 16.13.4, using the local homogeneous Dirichlet problem and the capacity theory of section 7.1. Since potentials with minimal singularities are locally bounded outside the pole set of a reference potential with analytic singularities, the argument can be carried out in charts where they are bounded; it uses no differentiability of the envelope.

The concentration theorem of that source also allows pseudo-effective classes and quasi-continuous obstacles that are not bounded above. Obstacles that are not bounded above are already covered by theorem 16.13.4, because capping the obstacle strictly above the envelope changes neither the envelope nor its contact set. Pseudo-effective classes are treated in corollary 17.5.38, after the criterion for bigness: if the class is not big, every non-pluripolar measure of top degree vanishes.

For functions whose Laplacian with respect to a metric is bounded, and for the reduction to regular contact, compare [p. 4 and Lemma 3.1, pp. 4–5, DNT2019v1]. Such functions are locally in $W^{2,2}$ by lemma 16.14.4, which is derived from the elliptic theory of lemma 10.2.5. Proposition 16.14.6 applies the vanishing of weak derivatives on level sets to obstacles in $W^{2,1}_{\mathrm{loc}}$, for instance continuous obstacles with bounded metric Laplacian; the computation on the contact set needs only this local regularity. The stronger elliptic $W^{2,p}$ estimate quoted in the source is proved separately in proposition 16.14.9, by freezing the principal coefficients and testing the equation against a regularization of $u|u|^{p-2}$, and corollary 16.14.11 then shows that the signed determinant of such obstacles is integrable on all of $X$.

Approximation of envelopes by Monge–Ampère equations with an exponential penalty, and the limit as the parameter tends to infinity, are due to Berman; the equations and estimates are presented in [equations (2.1)–(2.2), pp. 282–283, Tos18], where smooth solvability is referred to Aubin and Yau. We prove solvability for positive parameters in theorem 16.15.20 by the continuity method, with a contraction argument in Sobolev spaces and a priori estimates along the path. The third-order estimate, lemma 16.15.13, bounds the difference of two Chern connections. Its computation follows the tensor calculation in [Proposition 2.8 and Theorem 2.9, pp. 24–26, SongWeinkove2012v1], the first arXiv version of Song and Weinkove's lecture notes, where such estimates are traced back to a calculation of Calabi and the explicit formula in the parabolic case is attributed to Phong, Šešum and Sturm. Our calculation is elliptic: the penalized equation bounds the Ricci tensor and its covariant derivative in terms of the difference of connections.

The constants in the uniform trace bound and in the convergence to the envelope (lemma 16.15.7, proposition 16.15.21) depend only on the fixed geometry, on a bound for the obstacle $f$ and on an upper bound for the trace $\Lambda \mathrm{dd}^{\mathrm{c}}f=-Lf/(4\pi )$ of its complex Hessian, that is, on a lower bound for its metric Laplacian $Lf$. Tosatti writes forms with $\mathrm i\partial \bar\partial $; in the notation of section 16.15, the coefficient matrices $h$ and $k$ represent $2\pi \omega $ and $2\pi \kappa $, so that $g_{i\bar j}=k_{i\bar j}+a_{i\bar j}$ is the coefficient matrix of $2\pi (\kappa +\mathrm{dd}^{\mathrm{c}}a)$.

That the envelope of the minimum of finitely many functions with bounded metric Laplacian again has bounded metric Laplacian is [Theorem 2.5(ii), p. 5, DarvasRubinstein2014v1] (first arXiv version). We prove it in theorem 16.15.24 by smoothing the finite minima and applying the uniform trace estimate above; in the trace of the complex Hessian of the smoothed minimum, the variance of the gradients enters with a negative sign, so the upper bound for this trace is preserved. Convolution in charts and the elliptic estimates for all finite $p$ (proposition 16.14.9) control the commutator of the convolution with the operator $\Lambda \mathrm{dd}^{\mathrm{c}}$, whose coefficients vary.

The estimate for the full real Hessian (lemma 16.16.1, theorem 16.16.7) follows Tosatti's argument with penalized equations, [pp. 283–287, Tos18]. For the absorption of the negative term with third derivatives, Tosatti refers to the calculation of Chu, Tosatti and Weinkove, which is given in full in [arXiv version, Section 2, Lemmas 2.1–2.2, pp. 4–10, CTW17]. In this book the formulas for derivatives of eigenvalues and every absorption step are proved directly. The perturbation by a projection subtracts one from the eigenvalues other than the largest one; keeping track of this shift gives the bound $2\lambda +3$ in the comparison of positive forms. The real metric associated with $2\pi \kappa $ is $2\operatorname{Re}(k_{i\bar j}\, \mathrm dz_ i\otimes \mathrm d\bar z_ j)$, so that its Hessian trace is $2D_ ka$. Since the estimate uses, besides bounds for the obstacle and its gradient, only an upper bound for its real Hessian, the same proof applies to minima of finitely many smooth functions, and then to rooftop envelopes of finitely many $C^{1,1}$ functions.

The regularity of envelopes in big classes on the ample locus is [Theorem 1.1, pp. 1347–1349, DNT23]; this locator refers to the statement in the published article. The complete proof and the extension to finitely many obstacles are given in [arXiv version, Lemmas 3.1, 3.3–3.5, Section 4 and Theorem 4.2, pp. 3–19, DNT23]. The smoothing by the flow of a Chern connection goes back to [Proposition 2.9, pp. 5–6, Proposition 3.8, pp. 8–12, and Theorem 4.1, pp. 12–16, DemFlowOnline], a twenty-page manuscript cited with its own page numbers. In lemma 16.17.1 we carry out the computation with jets of a fixed finite order instead of an extension to infinite order, and compute the integrals of the weights directly. Demailly's normalization $\mathrm i\partial \bar\partial /\pi $ is twice our $\mathrm{dd}^{\mathrm{c}}$, while the auxiliary operator in our kernel computation is $\mathcal D=2\pi \mathrm{dd}^{\mathrm{c}}$.

Theorem 16.19.1 follows the two successive smoothing estimates of Di Nezza and Trapani and their use of high powers of the equations of the divisors on a resolution. Local holomorphic changes of the radius give the required uniform estimates without introducing the total space of the line bundle used by them, and lemma 16.18.1 handles the minimum over varying radii at the upper end point. We keep a fixed quadratic correction in the smoothing, keep the smooth background separate from the nonsmooth obstacle, and recover the varying metric Laplacian from the quadratic error by an expansion of the adjoint kernel. The final local estimate in real Sobolev spaces follows from the Euclidean Hessian multipliers of section 2.9.

The formula for the Monge–Ampère measure on the contact set for arbitrary singular potentials is [Theorem 3.2 and its proof, pp. 5–8, DNT2019v1], where the method is credited to earlier work of Witt Nyström. In theorem 16.20.2 we simplify the passage to the limit in the weights at levels: the limiting inequality is restricted to the contact set, where the positive weight cancels, so that a single truncation level and a single positive parameter suffice. For a general obstacle, whose metric Laplacian need not control its full Hessian, we keep the regular intermediate envelope.

Corollary 16.20.4 gives the formula of the source for the density on the contact set of an ordinary obstacle in big classes. Corollary 17.5.39, proved after the criterion for bigness, also covers pseudo-effective classes that are not big, in which all non-pluripolar measures of top degree vanish. All measures are taken with our normalization; with the operator $\mathrm i\partial \bar\partial $ of the source, products of $n$ factors, pure or mixed, are $(2\pi )^ n$ times ours.

The argument showing that envelopes of singularity types preserve mixed masses is in [Section 3B, p. 2060, DDNL18mono]; the continuity along increasing sequences that it requires is provided by corollary 13.8.9, proposition 16.21.2. The bounds for the Monge–Ampère measures of rooftop envelopes and of envelopes of singularity types are [Lemma 3.7 and Theorem 3.8, pp. 2060–2061, DDNL18mono]. In lemma 16.13.1, corollary 16.13.5, proposition 16.21.3 they are derived from convergence within a bounded singularity type and from the theorem on contact sets for obstacles. Here a measure is carried by the contact set if the complement of this Borel set has measure zero, which says nothing about the closed support of the measure.

The extrapolation argument and the domination principle (theorem 16.22.4, theorem 16.22.10) follow [Lemma 2.4.4 and Theorem 2.4.5, XiaSampleBook]. In lemma 16.22.1, a fixed point that preserves the mass is obtained as the regularized limit of an increasing sequence that almost maximizes an integral height at every step; this replaces the transfinite iteration in the proof in the source. For theorem 16.22.11, proposition 16.22.14 compare [Theorems 3.12 and 3.14, pp. 2063–2064, DDNL18mono]. There the domination principle is derived from the construction with prescribed densities in Corollary 3.9, p. 2062, whereas here, in a big class, it follows from extrapolation and the concentration on contact sets. The example with a $\log \log $ singularity, example 16.22.12, follows [Example 3.1.1, XiaSampleBook]; its mass is computed with the identity in degree one, proposition 13.4.7.

The criterion for the existence of rooftop envelopes, the substitution of potentials of full mass and the mass inequality (proposition 16.23.1, proposition 16.23.3, theorem 16.23.5) follow [Proposition 3.1.5, Lemma 3.1.2 and Theorem 3.1.3, XiaSampleBook]. The bound for upper limits needed there is isolated in lemma 16.23.4; it does not claim weak convergence of the measures along a decreasing sequence, which may lose mass.

Envelopes with an obstacle and a constraint on the singularity type (definition 16.25.1, proposition 16.25.2, proposition 16.25.3) extend [Definition 3.1.4, XiaSampleBook]. The bound on the contact set for bounded quasi-continuous obstacles follows from the theorems on relative envelopes and on concentration for obstacles; it does not identify a smooth density. Proposition 16.25.4 is [Proposition 3.1.15, XiaSampleBook], with an explicit defect term for potentials that do not have full mass. The relative capacity of definition 16.25.5, defined with competitors between $\phi -1$ and $\phi $, reduces to [Definition 2.3.1, XiaSampleBook] when $\phi $ has minimal singularities. Theorem 16.26.1 gives a complete proof of [Theorem 6.1.1, XiaSampleBook]; in big classes it provides the margin for rooftop envelopes used in [Theorem 3.1.1, VuXiaNakamaye].

Our definition of model potentials keeps the condition of positive mass from [Definition 3.1.3, XiaSampleBook]. For the constraints by multiplier ideals used in the chapter on envelopes of that book, proposition 16.24.1 combines the uniform valuative criterion theorem 6.12.16 with lemma 16.24.2, the formula $\nu (v,E)=\inf _ j\nu (v_ j,E)$ for the Lelong numbers along divisors of an upper envelope $v=(\sup _ jv_ j)^*$. In the proof of this formula, a local pullback argument shows that these Lelong numbers are preserved under the countable selection of Choquet's lemma. The proposition thus provides the closure property used in [Propositions 3.2.6 and 3.2.9, XiaSampleBook]; the construction itself and its algebraic consequences are not treated here.

The comparison of capacities in section 16.27 follows Lu [arXiv version, Lemmas 3.1–3.4 and Theorem 3.5, pp. 8–13, Lu21]. Integration by parts for bounded potentials on a compact manifold, lemma 13.5.2, is derived from the calculus of gradients in section 7.14, and, together with approximation by smooth bounded potentials, it gives the weighted estimate lemma 16.27.2. The envelopes with increasing penalties of lemma 16.27.5 provide the concentration on compact sets and the two growth bounds, which replace the properties of relative extremal functions quoted in the preprint. The comparison then proceeds in three steps: powers of model potentials in Kähler classes (proposition 16.27.12), comparison downwards for potentials of full mass (lemma 16.27.15), and comparison in both directions (theorem 16.27.17). Finally, proposition 16.27.18 compares, by a local extension construction, these capacities with the local capacities of section 7.1; in particular, local convergence in capacity implies convergence in each relative capacity.