25.15 Singularity classes and traces81F92F228A7E

81F92F228A7E

The countable-pole construction of theorem 18.13.1 uses the local decreasing-limit and canonical-representative theorems in Demailly's book [Chapter I, Theorems 5.4 and 5.8, pp. 39–40, DemCADG]. Its exponential integrability follows directly from weighted Jensen's inequality and the sublinear logarithmic profiles. The envelope argument of proposition 18.13.4 uses Choquet's lemma and the local upper-envelope theorem [Chapter I, Lemma 4.23 and Theorem 5.7, pp. 38–40, DemCADG], through the exact nonclosed-background construction of section 5.40. The bounded unrestricted envelope has the same trivial multiplier ideals as the constructed potential, so it is a competitor for the ideal-equivalence envelope used here. The fixed stronger potential supplies ordinary competitors below every continuous obstacle; the general zero-obstacle implication remains open.

A theory of singularities for nonclosed backgrounds must record torsion as well as divisorial information. The quotient of the space of smooth real $(1,1)$-forms by the forms $\mathrm{dd}^{\mathrm{c}}f$, $f$ smooth, used in section 10.30, appears in [Section 1, BGL25]; fixing the exterior derivative of the form keeps this datum on each modification.

Resolution makes additivity of traces accessible to approximation by multiplier ideals. The comparison in theorem 20.3.4 is the argument underlying Cao's additivity theorem [proof of Theorem 1.4, Cao15], and it does not require a closed global background. The recovery theorem theorem 13.21.7 combines the arguments with decreasing models and stretching with the local criterion for multiplier ideals. Uniform exponential integrability bounds each local divisorial order of the auxiliary potentials, so the proof does not require a local modification to extend over the compact manifold. The construction on a ruled surface in subsection 16.29.15, based on Demailly, Peternell and Schneider's [Example 1.7, DPS94], exhibits the boundary behavior relevant to this recovery.

The transverse recovery of theorem 16.29.22 uses the same positive-volume theorem on $\mathbb P^2$, while the Hopf threefold itself has unbounded upper mass by theorem 23.1.1. The full ideal envelope is invariant under the elliptic action because its complete competitor family is invariant; a horizontal current test then identifies it with the base envelope. The nonanalytic family uses the summable divisorial criterion of 13.24.8, whose closed mass and valuative inputs are described above. The general exact-pullback identities in proposition 16.28.59, proposition 18.4.3 follow from the descent, ideal transform and non-pluripolar product foundations; they do not assert recovery over arbitrary new Hermitian perturbations on a modified manifold.

The continuity of model envelopes in closed classes used in theorem 20.3.33 is due to Darvas and Xia [Proposition 2.1, DX24]; it is also presented in [Proposition 3.1.12, SGPT]. The argument for Lelong numbers uses small convex corrections from Darvas, Di Nezza and Lu [Lemma 4.3, DDNL21]. It suffices to apply this continuity on the submanifold, because the restricted approximants have a common curvature bound even when the ambient background is not closed.

The projective-base construction of subsection 20.3.41 uses the same positive-mass convergence of decreasing closed models; the October 4, 2026 version of the lecture manuscript gives the statement and proof in [Proposition 3.1.12, pp. 96–97, with Lemma 2.4.4, pp. 67–71, SGPT26Oct]. The ideal-class argument uses its compact Kähler criterion [Proposition 3.2.1, pp. 122–123, SGPT26Oct]; the cofinality discussion on p. 497 covers prime divisors on arbitrary compact smooth models by finite smooth blowup models. These are closed-base inputs. The finite extraction proof in lemma 20.3.44 uses proper mapping for analytic sets and analytic resolution which preserves the whole nonsingular locus, in the form of Włodarczyk's [Theorem 2.0.1(1), p. 34, Wlo09]. Its ramification argument uses the local Siu decomposition and Skoda integrability recalled in the foundations. Elliptic symmetry gives the full Hopf envelope and its mass and approximation formulas; the ideal-class argument remains on the projective base.

The residual condition in (18.27.1) implies vanishing of point Lelong numbers by scaling the uniform Lelong bound, and vanishing of polar curvature by the Gauduchon trace and bounded-type invariance. The argument persists on every smooth compact modification. The ideal neutrality of lemma 5.37.5 combines Bergman approximation and the local ideal-addition calculus; the needed Bergman estimate and strong openness are stated in [v2, Theorem 1.2 and Corollary 2.3, DemCohomology15]. The smoothing step in lemma 18.27.3 is Demailly's [Main Theorem 1.1 and Section 2, Dem92], applied to the closed current $T=\mathrm{dd}^{\mathrm{c}}q'$ with smooth representative zero and lower bound $\epsilon h-B$. In the source normalization the potential is $q'/2$. The curvature of a fixed tangent metric is bounded by a large multiple of $h$, and its continuous Lelong approximants converge uniformly to zero. Richberg smoothing is applied to the continuous attenuation intermediates. This operation supplies a smooth positive residual background; the prescribed ordinary type is recovered with an actual small-curvature representative in theorem 18.27.4. The normal crossing ideal formula in proposition 20.3.8 uses ambient ideals and therefore remains meaningful when the naive residual restriction has poles everywhere on the target.

The full ordinary-type optimization in theorem 20.3.66 uses the closed comparison theorem of Boucksom, Eyssidieux, Guedj and Zeriahi [Theorem 1.16, pp. 217–218, BEGZ10] in both orders, in degrees one and two. Their Definition 1.2, p. 204, applies because every residual potential is locally bounded outside the divisor. Finiteness and closedness of the auxiliary non-pluripolar products are their [Proposition 1.6 and Theorem 1.8, pp. 206–209, BEGZ10]. Only the dominating Kähler background enters this cohomological comparison; the nonclosed mass is recovered by smooth-background expansion and distributional Stokes. The two ordinary volumes and their divisor and pure-pullback laws are those of [Definition 3.1 and Proposition 3.3, BGL25] in their bounded-mass setting; here the latter identities are proved through the complete competitor correspondence, and the small-curvature sublogarithmic representatives give the extreme sequences. The ambient ideal orders of proposition 20.3.71 follow from a transverse radial integral. For proposition 20.3.74, the small-curvature criterion first produces a smooth residual representative; restriction to the last exceptional curve then excludes ordinary-type strictness even over pole centers. The exceptional curvature perturbation constructs a one-sided strict witness inductively. This extends the bounded-center restriction mechanism in example 10.17.9 to the specified finite blowup family. The separate residual volume and cost in 12.35.8 use the all-test dual formula and first variation of this book. The logarithmic degeneration in example 12.35.13 specializes those formulas to the height distribution on the sphere; pairing against a closed base test excludes every strict current at the endpoint.