25.4 Mass and energyBB36F353AA0C

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On a nonclosed background, the variation of the Monge–Ampère mass and the failure of the usual energy to have a primitive are related manifestations of torsion. Chiose's polarization argument [Theorems 1–2, Chiose16] gives the characterization of mass invariance in theorem 11.1.7. Yi Li constructed energies corrected by torsion terms [Equation (1.18) and Theorem 1.2, YiLi10]. The correction changes the one-form being integrated, in agreement with theorem 11.2.4, which shows, for $n\geq 2$, that the uncorrected one-form has no primitive unless $\omega $ is closed; proposition 11.2.10 computes the difference by a first variation.

Cutoffs of diffuse poles separate this variation of the background from a fixed loss of mass at the singularity. Let $n\geq 2$, and let $H$, $C$ and $M$ be the cutoff, the curvature constant and the residual mass of proposition 18.26.19; fix $0<t<1/C$ and put $A=\int _ X\omega ^ n$. The mass of $tH$ is a polynomial in $t$ whose finite torsion coefficients come from Stokes calculations for compactly supported bounded products. By proposition 18.26.21, every representative that is ordinarily bounded-equivalent to $tH$ loses the same top residual mass $Mt^ n$, although its lower-degree coefficients can vary. If $\mathrm{dd}^{\mathrm{c}}(\omega ^ q)=0$ for $1\leq q<n$, invariance of the mass under bounded changes and invariance of the residual give the mass $A-Mt^ n$ throughout the class (corollary 18.26.20); this applies to every compact complex surface with a Gauduchon form. On a surface, corollary 18.26.22 shows that the mass is constant on the class if and only if $\omega $ is Gauduchon, since admissible smooth bounded perturbations detect any failure of the Gauduchon condition.

The failure of the bounded mass property on a Hopf threefold is the counterexample of Xia and Zhang [Theorem 1.1, XZ26]. There a Gauduchon background converts vertical Dirichlet energy into Monge–Ampère mass, a mechanism developed in section 23.1. Combined with theorem 13.14.18, it produces infinite extended mass with trivial multiplier ideals at every positive weight, and the results on integrability and convergence of coefficients in subsection 13.14.20 describe the linear current carried by this construction. Xia and Zhang use the operator $\mathrm i\partial \bar\partial $ [Section 1, XZ26]; dividing their background forms by $2\pi $ and leaving their potentials unchanged preserves positivity and changes the coefficient of the mass identity.

The local matrix, heat potential, Gaussian gradient estimate and completed square used in subsection 23.1.41 are those of Xia and Zhang [v1, Lemma 2.2, pp. 4–5; Lemmas 3.1–3.3, pp. 6–9; equation (4.2), p. 10, XZ26]. The faster initial time in theorem 23.1.52 changes the radial length and the integrated defect, so its positive correction is proved at that scale. The positive fiber flux and depth estimate give the square-root upper bound for all potentials in the declared unitary invariant class. The lower-density regions then give a single infinite weighted measure and concentration on a pole divisor at the critical weight; they also impose necessary conditions without the symmetry assumption. The unrestricted optimal sufficient rate remains open.

The passage from smooth to bounded and unbounded invariant potentials uses the classical bounded product and capacity theory developed in section 5.27, section 5.29. The local product construction and decreasing continuity are Bedford and Taylor's [Theorem 2.1, pp. 4–7; Theorem 2.4, pp. 7–9; Corollary 2.8, p. 10, BT82]; their quasi-continuity and zero-capacity results are [Theorem 3.5, p. 14; Theorems 6.9–6.10, pp. 25–26, BT82]. They use $\mathrm{dd}^{\mathrm{c}}=2\mathrm i\partial \bar\partial $, which is $4\pi $ times our operator and changes capacities only by a fixed factor. The local $L^1$ compactness underlying theorem 5.25.2 is also described in [Chapter I, Proposition 4.21, pp. 36–37; Proposition 5.9, p. 40, DemCADG]. In the present argument, group convolution and canonical truncation give the original Hermitian capacity convergence, including the boundary mass at a truncation level.

The extension in theorem 23.1.35 keeps the same threefold heat family. Its proof first bounds the actual mass on the negative radial half-line by exact integration over the elliptic fiber, then uses lemma 23.1.33 to compare the angularly integrated densities in different dimensions. The additional angular factor has a fixed positive lower bound on the half-line carrying the divergent mass. Corollary 23.1.36 transfers this failure to every Hermitian metric on the same scalar Hopf manifold.

The failure of positive lower mass is a different phenomenon; it occurs on the elliptic threefold of section 12.19. Sufficient conditions for a positive lower bound come instead from the sign of the torsion: (12.16.4) is the three-dimensional case of a formula of Angella, Guedj and Lu [Theorem 3.2, AGL23], who call the condition $\mathrm{dd}^{\mathrm{c}}\omega \leq 0$ plurinegativity.

The principal elliptic bundles of subsection 24.4.7 are special cases of the torus bundles of Goldstein and Prokushkin [Section 2, GP04]. The metric calculation on these bundles gives an upper bound for the mass, but it does not decide whether the lower mass vanishes. The radial calculation of proposition 24.4.13 bounds the root volume of the specified affine test family at fixed degree; the lower mass over all potentials remains open.

The completion energy of subsection 11.2.14 is the Dinew–Popovici functional. Hanwen Liu defines it and computes its variations in [v1, Section 2.1, pp. 3–5, LiuCritical26], and proves the identity relating energy and volume along potential variations, together with the indefiniteness of the Hessian, in [Theorems 5.1–5.2, pp. 16–19, LiuCritical26]. Liu assumes that the ambient manifold is Kähler. The proofs in the book use only the finite-dimensional space of closed holomorphic completions and Stokes' theorem, so these statements about potential variations hold on every compact Hermitian symplectic fourfold. Liu's calculation of criticality for products [Lemma 2.3 and Theorem 2.4, pp. 7–9, LiuCritical26] gives the mechanism of example 11.2.23, where an explicit Fourier form on the square torus replaces the general existence argument on surfaces. Neither the general existence criterion of that paper in terms of the canonical bundle nor its convergence theorem for the global torsion flow is used here. Liu's convention is $\mathrm d^{\mathrm c}=\mathrm i(\bar\partial -\partial )$; rewritten with the book's $\mathrm d^{\mathrm c}$ and $\mathrm{dd}^{\mathrm{c}}$, the formulas of that paper for potential variations give (11.2.20) with the coefficients shown there. Combined with Chiose's criterion, the energy identity gives the equivalence between flatness of the energy and invariance of the mass in corollary 11.2.21. The calculation with one nonclosed factor in example 11.2.24 shows that this flatness along potentials implies neither criticality in the full Aeppli class nor the existence of a primitive for the Monge–Ampère one-form.