25.13 Normalization and adjoint measures7B9538DE7653

7B9538DE7653

The normalization constant can be studied through its adjoint probability measures even when the potential itself is not unique. The differential method for smooth solutions appears in Collins's work [Collins21]; theorem 11.33.4 replaces differentiation of smooth solutions by mixed products and domination.

The continuous-Hessian extension in theorem 11.34.11 uses local bounded Bedford–Taylor continuity and the finite background expansion to linearize the equation along signed potential variations. The supporting-density estimate of corollary 12.9.3 then allows Sobolev duality with lemma 11.24.1; the directional derivative is the convex-analytic identity already proved in proposition 11.34.5. The product formula in proposition 11.34.13 uses curve Green inversion and the smooth Gauduchon multiplier on the other factor. Its two-curve specialization is also covered by theorem 11.45.16 with zero divisor. The non-Dini example uses the same Hopf torsion calculation as (23.1.3), in complex dimension two.

For the mass quotient and the closed examples in section 11.42, existence and uniqueness of solutions in a prescribed singularity type are due to Darvas, Di Nezza and Lu [Theorems 1.4(i) and 4.29, pp. 2052 and 2079–2080, DDNL18mono], and the invariance of the mass under bounded changes of the potential is their [Proposition 2.1, p. 2055, DDNL18mono]. Their hypotheses are a compact Kähler manifold, a smooth closed big class, a model reference with small unbounded locus, and positive mass. The factorization with positive residual concerns the ordinary divisorial singularity types themselves; it does not assert an identity of model envelopes for every weighted sum with real coefficients.

The dictionary between toric potentials and convex functions used in section 11.43 is that of Berman and Berndtsson [arXiv v1, Sections 2.1–2.3, especially Proposition 2.3 and Lemma 2.5, pp. 10–13, BBToric13]. The angular coefficient in (11.43.4) is computed directly, in the book's conventions for the squared logarithm and the operator. The argument with finite envelopes establishes the identity for ordinary models, and a common translated simplex gives both the witness in a smaller background and the volume estimate in terms of the Hausdorff distance. The two families in subsection 11.43.9, subsection 11.43.14 satisfy these conditions explicitly; since their references have small unbounded locus, the theorem for prescribed singularity types in closed classes cited above applies. The blowup chart and the normal line of the exceptional divisor used to compute the rank and the mass of the residual are taken from [Chapter VII, equation (12.2) and Proposition 12.4, pp. 356–357, DemCADG]. These computations concern fixed modifications and closed backgrounds.

The description of the residual in proposition 11.34.18 uses the products of bounded plurisubharmonic functions of Bedford and Taylor [Section 2, BT82] and the extension of plurisubharmonic functions across pluripolar sets in [Chapter I, Theorem 5.24, DemCADG]. Integration by parts for bounded potentials against a closed positive current is established in [Theorem 1.14 and Lemma 1.15, pp. 214–217, BEGZ10]; alternatively, the cutoff in (11.34.20) can be smoothed explicitly, which makes its estimate a direct Stokes calculation for distributions. In theorem 11.34.21, the two radial profiles give the coefficients of the divisors before the limit (11.34.23) of the squares is taken. This distinguishes the relative residual adjoint from the classical adjoints of smooth potentials whose exponential densities concentrate.

For the normalization on split products in lemma 15.30.2, the smooth input is again [Corollary 1, TW10], applied to the closed projective factor. The order of the scalars for bounded densities extends the formula to continuous densities on the factor, and the original truncations determine the threshold for every smooth reference volume. The radial flux in lemma 15.30.1 and a separate calculation of non-pluripolar products then give theorem 15.30.3. The elliptic factor is the threefold of section 12.19, whose balanced metric has volume $24\pi $ by proposition 12.19.18; this gives the fivefold equation with constant $240\pi $. The bounded equation of [Theorem 0.1, KN15] and closed Stokes extend this result to every integrable density from a closed Kähler factor in theorem 15.30.6: comparison of the original truncations computes the scalar, and threshold attainment on the factor, together with its non-pluripolar mass bound, makes the lower measure an equality. The root norm formula of proposition 15.32.11 then uses the full Gauduchon test class and identifies a closed subspace of $\mathcal D_ h^0$. The radial family supplies roots outside every higher $L^ r$ class, with explicit logarithmic tails. For arbitrary densities on the product that are merely integrable and have positive threshold, the existence of an exact solution remains the problem of 24.2.2.

The same radial flux gives the criterion for exponential growth in theorem 15.35.9. Disjoint annuli in projective space and a rapidly convergent smooth baseline adapt the concentration in theorem 15.35.10 to every prescribed sequence of base densities whose scalars collapse. The bounded equations on the products determine the scalars of the components, and for every positive exponent the exponentially weighted pole absorbs their sum. The construction of greatest solutions then gives exact solutions at zero threshold, which are necessarily unbounded, in particular on the elliptic products of dimensions four and five. The explicit barriers there have masses $96\pi $ and $240\pi $, whereas the exact solutions have their own finite positive masses. A general criterion for solvability at zero threshold remains open.

That the set of solvable exponents is an upper interval, proposition 15.35.12, follows by translating a subsolution downward before increasing its exponent. For the collapse of mass at zero threshold in proposition 15.35.13, normalized exponential damping gives an $L^1$ approximation of the fixed density, and upper semicontinuity of the threshold forces the normalization scalars to tend to zero. This controls the masses of the exact solutions without identifying the limits of the normalized potentials.

The criterion for attaining a positive endpoint, proposition 15.35.16, combines the compactness of normalized potentials with the closure of subsolutions under lower bounds for the density and the existence of greatest solutions. Its common upper barrier comes from the equation of theorem 14.19.3 with the truncated, bounded density, together with singular strict domination. The corrected order in lemma 15.35.15 gives a decreasing family as the exponent decreases to a positive solvable exponent. Monotone convergence in capacity and the closedness theorem for positive exponents identify the limit from the right in proposition 15.35.17, and maximality of the greatest solutions identifies the limiting potential. Approach from below uses the regularized increasing envelope, giving the exact left limit of proposition 15.35.18. Integrating the corrected family gives a bounded increasing real function on each compact exponent interval. Its jumps detect the difference between the left limit and the greatest solution, which proves proposition 15.35.19.

For the non-$L^ p$ radial family, lemma 15.36.2 identifies the greatest solutions on the full product through the bounded factor equations and singular strict domination. The proof of theorem 15.36.5 uses their cumulative fluxes to determine the whole normalized limit, including when the density parameter varies. Local uniform convergence outside the fixed pole set and the Hermitian capacity estimate give capacity convergence on both factors and the product. No general full-mass stability theorem is needed for this radial argument. The finite or divergent first moment determines scalar centering in proposition 15.36.8. The order characterization of separated critical subsolutions in corollary 15.36.11 additionally uses Dinew's closed-class full-mass uniqueness theorem [arXiv v1, Theorem 1.2, pp. 2 and 5–8, DinewUniqueness08]; this input is applied only on the projective factor. It neither proves uniqueness of arbitrary nonclosed critical potentials nor supplies a theorem for varying singular references.

For the logarithmic product references, proposition 11.45.3 computes the ordinary envelope on the curve and identifies the product envelope by restriction to slices. Divisorial removal transfers the exponential equation and its greatestness to a bounded equation in the positive residual background, as in lemma 11.45.4. The Green operator on the curve gives uniformly bounded residuals by contraction for small exponents. Its uniformly integrable logarithmic kernel then determines the joint limit and the scalar error in theorem 11.45.9. The centering uses the critical scalar of each varying datum. The separated order characterization in corollary 11.45.13 follows from closed curve mass and the distributional equation; maximality of the coefficient in the full prescribed class follows instead from greatestness of the positive-exponent solutions. This argument uses the earlier bounded existence and domination theorems and does not impose a mass property on the Hermitian factor.

For the product surface in 11.45.14, the logarithmic model and Green residual are those of proposition 11.45.3, theorem 11.45.9. The continuous positive residual allows smooth perturbations of the singular solution, and the supporting inequality on all continuous data makes every supporting probability invariant for the weak linearization. Spherical harmonics reduce uniqueness to the signed-measure resolvent lemma lemma 11.45.15, whose proof uses the round sphere heat operator. This gives theorem 11.45.16 without an invariant-density regularity assumption. The approximation statement proposition 11.45.18 uses the already proved conformal Gauduchon theorem theorem 11.4.2 for each smooth residual; uniform tensor convergence transfers the operator and then the weighted products. The explicit pole-filling family in example 11.45.19 shows why retaining the divisor is part of this approximation statement.

The one-sided divisorial correspondence of lemma 11.41.5 extends the ordinary correspondence of theorem 11.41.1 by the nonclosed divisorial-removal lemma and locality of the extended measures. A uniform submean bound identifies the ordinary envelope type and its explicit smaller-background competitor. The whole one-sided threshold in theorem 11.41.6 then follows from theorem 15.28.4; its smooth attainment uses theorem 14.2.1, rather than a new prescribed-singularity existence theorem. Strictness follows from the elliptic cofactor identity off the divisor and the strong maximum principle. The obstruction in corollary 11.41.7 is consistent with the stronger later smooth-dual rigidity, while example 11.41.8 fixes the Fubini–Study factor to compute the precise restricted coefficient. The inverse realization problem for general densities and annihilating faces remains separate.

The curve-measure equations of section 15.31 begin with the normalized logarithmic kernel on the projective line. Its distributional curvature and the degree-one extended measure give proposition 15.31.13 by bounded product expansion and truncation. Under continuity of the potential, local Riesz decomposition gives non-polarity, while rotation smoothing and Dirichlet energy give the canonical Sobolev trace of lemma 15.31.4. The convex functional in theorem 15.31.8 yields every positive exponent; the bounded Green density and uniform logarithmic tails give continuity, and fixed barriers give uniform selection. The full-product greatestness and coefficient in theorem 15.31.15 use the earlier bounded strict domination and ordered-contact theorems. The latter retains the equality-set term when an unbounded competitor is floored by a bounded solution. The Frostman test includes nonconstant weighted circles. The zero-lower-mass conclusion assumes the factor's vanishing lower mass; the general individual-measure classification is not a consequence of this product construction.

The radial residual criterion in section 11.46 uses central mass rather than an ordinary-model envelope identity. The local decreasing-limit property gives the genuine pole, while the radial determinant flux identifies its non-pluripolar probability and inverse. The central density error controls the flux and its first derivative without differentiation of that error, giving theorem 11.46.8. Flat errors, mass-preserving sharp examples and an open zero annulus follow from the same inverse in proposition 11.46.15, proposition 11.46.17, proposition 11.46.20. The conditional nonclosed product of corollary 11.46.22 uses bounded multilinearity and pullback by a projection, as proved in proposition 5.28.2, lemma 5.28.4, followed by the actual non-pluripolar truncation limit. It assumes zero lower mass of the factor and makes no assertion about arbitrary nonsymmetric equations.

The singular-obstacle envelopes of section 11.47 use the same sublogarithmic reference as the preceding radial density criterion. Unitary invariance reduces the full envelope to a convex minorant, and supporting tangents control both ends. On the right tail, the weighted slope and second-derivative estimates in (11.47.12) give uniform projective Hessian bounds even when contact moves to infinity; interpolation with value contraction gives theorem 11.47.11. The full product formula in proposition 11.47.14 follows from exact slicing of every competitor and the bounded-product expansion before strict-superlevel truncation. It requires no mass property of the Hermitian factor. These results concern this everywhere singular obstacle and the residual in its prescribed ordinary type, while regularity for arbitrary types and compact obstacle sets remains open.