25.1.17 Relative energy and integration by parts889877FC2044

The identity for differences of bounded type, lemma 17.2.3, follows [arXiv version, Lemma 4.1, pp. 13–15, Lu21]. Its proof reduces to bounded potentials by truncation, first for a strictly positive background; an expansion in polynomials then removes this restriction. The general integration by parts formula, theorem 17.2.5, follows the argument by differentiation and polarization in [arXiv version, proof of Theorem 1.2, pp. 15–16, Lu21]. In his introduction, pp. 1–3, Lu credits the earlier integration by parts formula to Mingchen Xia. The general formula is stated in the introduction of [Theorem 1.1 and discussion, pp. 1–2, Xia2019v1] (first arXiv version), which also describes a proof by a projective construction; the proof followed here is Lu's later argument with capacities.

Relative energy classes and the relative energy are defined in [Definitions 3.1.5–3.1.6, XiaSampleBook], and the cocycle formula for potentials of bounded type, the bounds by tangents, concavity and continuity along decreasing sequences are [Lemmas 3.1.3–3.1.5 and Proposition 3.1.16, XiaSampleBook]. In proposition 17.3.4 the first variation of the energy along a segment is computed directly from the integral formula for the energy and the integration by parts theorem, and the cocycle formula follows by integrating it. In proposition 17.3.12, proposition 17.3.13 the energy is extended by means of the canonical truncations $\max \{ u,\phi -k\} $, and continuity is proved for decreasing nets, not only for sequences, the limit being the pointwise limit.

For theorem 17.3.14, lemma 17.3.18, proposition 17.3.19 compare [Proposition 3.1.17, Lemma 3.1.6, Corollary 3.1.6 and Proposition 3.1.18, XiaSampleBook]. In our proof, a uniform bound at finite levels for the integral of a potential against its own Monge–Ampère measure shows that finite energy by itself implies full mass.

Continuity of the energy along increasing nets, proposition 17.4.1, follows [Proposition 3.1.20, XiaSampleBook]; the uniform bound for the tails is (17.4.2). The proof of [Proposition 3.1.21, XiaSampleBook] uses an exponential Monge–Ampère equation; in proposition 17.4.7 we instead integrate the bound for the measures of rooftop envelopes on contact sets, corollary 16.13.5. This closure of finite energy under rooftop envelopes guarantees that the slices of Perron segments have finite energy. The closure under upper envelopes of families and the envelopes of mutual singularities follow [Propositions 3.1.23–3.1.24, XiaSampleBook].

The derivative of the energy along affine segments, proposition 17.4.9, also appears in [Lemma 3.1.11, XiaSampleBook]. Here it follows from the cocycle formula, which is polynomial in the parameter. The family of smooth potentials on $\mathbb P^1$ in example 17.4.11 shows that convergence in $L^1$ does not imply convergence of the energy.

The characterization of solutions as maximizers of a functional and the derivative of the energy of projections (theorem 17.5.5, proposition 17.5.4) follow the variational approach of [Theorem 4.1 and Lemma 4.2, pp. 207–209, BBGZ13], whose normalization of the operator agrees with ours. We prove the derivative of the energy of projections directly, with a model potential as reference, from the theorem on contact sets and the bounds of the energy by tangents.

For Monge–Ampère equations relative to a model potential, with prescribed measure and with an exponential factor, compare [arXiv version, Theorem 4.3, p. 20; Theorem 5.3 and Lemma 5.4, pp. 24–25, DDNL19log]. That source uses $\mathrm{i}\partial \bar\partial $; its potential corresponding to our $u$ is $u/(2\pi )$, and its exponential parameter is $2\pi \lambda $. We solve the prescribed-measure equation with bounded densities (theorem 17.5.6) and the exponential equation for every finite non-pluripolar measure (theorem 17.5.13) by compactness in the variational approach. The relative comparison and domination principles give proposition 17.5.7 for every finite non-pluripolar measure, when the reference has positive mass.

For the continuity of integrals against a fixed non-pluripolar measure compare [Theorem 3.10, pp. 203–205, BBGZ13]. Lemma 17.5.11 proves the case of uniformly bounded functions. The upper limit is controlled outside a pluripolar set; for the lower limit we use convex combinations of the tails with almost minimal $L^2$ norm for the given measure, which form a Cauchy sequence by the parallelogram identity. Applied to exponentials, the lemma allows the same variational construction for arbitrary finite non-pluripolar measures. The lower bound for limits in lemma 17.5.14 follows [arXiv version, Lemma 2.8, pp. 12–13, DDNL19log]; a fixed lower measure is restricted before weak limits are taken. The existence of solutions with prescribed measure, theorem 17.5.15, is obtained instead by letting the exponential parameter tend to zero; for the existence statement compare Theorem 4.7, p. 23, of that preprint.

Uniqueness for the relative equations is proved in [Theorem 4.29 and its proof, pp. 2079–2080, DDNL18mono] by adapting the argument of Kołodziej and Dinew based on the concentration of mass; in that section the model reference has positive mass and small unbounded locus. Our theorem 17.5.17 allows an arbitrary model reference of positive mass. It follows the same strategy, using the existence theorem, the comparison and domination principles for full mass and the lower bound for products with respect to a common measure (corollary 13.5.11); the domination principle turns inequalities almost everywhere for the common measure into pointwise inequalities.

The iteration with capacities in theorem 17.5.20, corollary 17.5.26 follows [arXiv version, Theorem 3.3 and its proof, pp. 14–15, DDNL19log]. Here the measure is assumed to be dominated by the power $1+\delta $ of the capacity, and an iteration with geometrically decreasing steps shows directly that the capacities of the relevant sublevel sets vanish beyond a finite level. The uniform bound for moments is corollary 10.3.2.

The estimates of volume and of measures with $L^ p$ densities by capacity (proposition 16.27.8, corollary 16.27.10) follow [Proposition 4.30, pp. 2081–2082, DDNL18mono], which assumes a small unbounded locus and uses uniform Skoda integrability; the formulation for arbitrary model potentials is [arXiv version, Lemma 3.9 and Proposition 3.10, p. 18, DDNL19log]. Here the prerequisites are the uniform exponential bound corollary 10.3.3 and the penalized envelopes of lemma 16.27.5. At fixed potentials, the background $2\pi \theta $ of the source gives measures and capacities that are $(2\pi )^ n$ times ours, while their ratios to the mass of the reference are unchanged.

The criterion that a class containing a current of positive non-pluripolar mass is big, which is used in [Proposition 2.4.1, XiaSampleBook], is [Proposition 1.22, p. 222, BEGZ10]. There the criterion is attributed to Boucksom; in the published article [Section 4.3, Theorem 4.7, pp. 1057–1061, Bou02b] the numbering differs from that citation, and the volume is defined through the determinant of the absolutely continuous part. Our proof of theorem 17.5.37 uses instead the Monge–Ampère equations of section 17.5, solved in the big classes obtained by adding $\varepsilon \{ \omega \} $ to the given class, the mixed trace estimate lemma 17.5.33 and Lamari's criterion. The proof also shows that a pseudo-effective class is big if and only if its volume is positive, without using an identity between the determinant volume and the non-pluripolar volume.

Lamari's duality criterion, lemma 17.5.28, is [Lemma 3.3 and its proof, pp. 275–276, Lamari1999]; the cohomological statement on annihilators is on p. 267. We prove the Kähler case by extending a positive linear functional with the Hahn–Banach argument, together with the scalar Poisson equation and the identity for the trace and the Hodge star proved earlier.

The pointwise trace estimate for smooth forms and the integrated Cauchy–Schwarz argument are [Lemmas 3.1–3.2 and their proofs, pp. 4–6, Popovici2014v1] (first arXiv version). Lemma 17.5.33 states the estimate for an arbitrary strictly positive form $\tau $ of bidegree $(n-1,n-1)$, and extends it, by local decreasing convolution and linearization of the determinant, to currents whose local potentials are locally bounded outside a closed set of Lebesgue measure zero; the first factor on its left-hand side is the full mass of $S\wedge \tau $, including any mass on the poles. The preprint writes $\mathrm i\partial \bar\partial $ for potentials, whereas our proof uses the normalization of this book throughout.

The inequality between mixed masses and the geometric mean of the masses, with its consequences for the concavity of roots and logarithms of masses, is [arXiv version, Theorem 6.1 and its proof, p. 26, DDNL19log]; it is the result referred to in [Theorem 2.4.1, XiaSampleBook]. Theorem 17.5.41 follows the proof in the source by relative Monge–Ampère equations. Because this proof uses the criterion for bigness, the theorem comes after that criterion; the case of vanishing masses is treated separately. At fixed potentials, the operator $\mathrm i\partial \bar\partial $ of the source multiplies all measures of top degree by $(2\pi )^ n$; by homogeneity this factor cancels in the inequality.

The identities for envelopes of arbitrary mass in theorem 17.6.5 are [Theorem 1.2, p. 4, PSWZ26], an arXiv preprint; they include potentials of zero mass, to which theorem 16.22.11 does not apply. Their proof there (Section 5, Theorem 5.1, pp. 37–42) uses a uniform bound for normalized tails and the recovery of volume for perturbations of positive mass; proposition 17.6.1 follows this approach. Monotonicity gives the limit directly, and the solutions with constant density that are needed are provided by theorem 17.5.6. Applying the limit to each member of the family $\mathcal F_ u$ that defines the ceiling proves the identity $(\sup _{v\in \mathcal F_ u}v)^*=P_\theta [u]$, without a separate formula for the regularization of the ceiling and without a second family of perturbations.

The source uses the operator $\mathrm{dd^ c}=\mathrm i\partial \bar\partial /\pi $ (p. 7). At fixed potentials, its backgrounds $2\theta $ and $2\omega $ give measures of top degree that are $2^ n$ times ours, while the normalized tail ratios and the constant densities of the auxiliary equations are unchanged.