24.9.2 Recovery of compatible divisor systems5A6D078648B3

Prescribing finitely many divisorial poles leaves a smooth residual background. When the residual has a smooth positive representative, theorem 12.35.4 solves the residual equation for a smooth strictly positive probability measure, and every subsolution at the same scalar whose Lelong numbers along the prescribed divisors are at least the prescribed coefficients differs from this solution by a constant. Consequently a solution in the prescribed ordinary type has the same scalar and a smooth remainder. Passing to compatible data on all modifications requires keeping this positivity and this exact singularity information uniformly.

Passing to all modifications raises two further issues. First, a smooth positive probability measure pulls back to a density that vanishes on the exceptional divisor, and the smooth dual formula cannot be applied to such a density until a degenerate extension has been proved. Second, the residual classes on two models are related explicitly: corollary 10.30.5 subtracts the Bott–Chern class of the effective exceptional divisor while keeping the prescribed exterior derivative. The approximating coefficients converge in linear trace mass, but this formula gives no transformation rule for the nonlinear volume. Exact pullback preserves singularity volumes by proposition 16.28.59; subtracting an exceptional divisorial class changes the background and is a different operation. Assigning an intersection number to a nonclosed form resolves neither the degenerate equation nor the question of volume.

The envelope systems of section 16.29 are compatible families, over the positive Hermitian perturbations of a background, that record representatives of one multiplier-ideal class (definition 16.29.6). Their construction gives an order adjunction and an exact criterion for recovery: the limit as the perturbation vanishes must keep the prescribed ideal type. Under bounded mass, for a big background, theorem 13.21.7 gives a sufficient condition involving only the initial slices: their lower singularity volumes must have a positive limit as the perturbation vanishes. Concavity then gives positive lower volume at every finite parameter, and stretching preserves all local divisorial orders. The ruled-surface example of that chapter has zero residual volume and shows why recovery fails without such a condition.

On the scalar Hopf threefold, theorem 16.29.22 gives a sufficient condition despite unbounded ambient upper mass. If every positive scalar slice is invariant under the elliptic fiber action and the initial transverse limiting model has positive lower volume on $\mathbb P^2$, the system is recovered at every finite parameter and every positive Hermitian perturbation. The symmetry constructs the full transverse system rather than assuming survival of the limiting ideals. Example 16.29.24 supplies nonanalytic ideal data with exact positive transverse volumes. The recovery identity and transverse volumes survive modifications in the exact pulled backgrounds, and smooth gauge transport preserves recovery. A criterion for general nonsymmetric systems and arbitrary new Hermitian backgrounds on modifications remains unresolved.

For a fixed finite divisorial support, proposition 12.35.7 settles the case where the residual stays in the positive cone: if the coefficients converge and the limiting residual background still has a smooth positive representative, the normalized smooth remainders converge smoothly, the singular solutions converge in capacity, their measures converge in total variation, and their logarithmic coefficients remain exact on every modification. Multiplier ideals can nevertheless jump at an integer coefficient. For divisors of sections moving in fixed finite-dimensional spaces, positive limiting weights and residuals give convergence of the full potentials (theorem 11.41.3), in each of the two regimes stated there. Recovery beyond the regime of bounded mass and positive volume, degeneration of residual positivity, and estimates uniform over infinitely many models all require further arguments.