24.9 Trace realization and recovery1B042D04CEDB

Prescribing divisorial lower bounds and taking an envelope may produce larger orders than those prescribed. Let $u$ be a quasi-plurisubharmonic function on $X$ and $Y\subset X$ a connected compact submanifold of positive dimension along which $u$ has zero generic Lelong number, $\nu (u,Y)=0$. As in section 20.3, the trace data $\tau _{Y,u}(E)$ of (20.3.3) are the asymptotic vanishing orders of the restrictions to $Y$ of the multiplier ideals $\mathcal I(tu)$, along prime divisors $E$ on smooth modifications $\pi \colon Z\to Y$. For a smooth real $(1,1)$-form $\Omega $ on $Y$, the trace envelope $\mathcal T_\Omega (u;Y)$ of (20.3.25) is the upper envelope of the nonpositive $\Omega $-plurisubharmonic functions $a$ with $\nu (a\circ \pi ,E)\geq \tau _{Y,u}(E)$ for all such $E$. The realization problem asks when these inequalities are equalities. Theorem 20.3.27 constructs the envelope in the restricted background itself and identifies it as the $L^1$ limit of the envelopes of the restricted analytic approximants, and theorem 20.3.30 shows that it pulls back exactly to every compact modification and is there the limit of the envelopes for vanishing Hermitian perturbations. When $Y$ has bounded mass and carries a Hermitian form of positive lower mass, corollary 13.21.5 proves equality in all divisorial inequalities for sufficiently large backgrounds $\Omega $; when $Y$ is Kähler, equality also follows directly from the closed theory (theorem 20.3.33). Ambient potentials with analytic singularities have exact ordinary restrictions. In higher dimension, theorem 20.3.39 realizes all orders when the data reduce to a problem on an ambient curve, with an ordinary type independent of a sufficiently large background. For general data, theorem 20.3.35 shows that in a larger Hermitian background the envelope has positive finite mass.

The equality is required on all modifications, which is stronger than realizing the divisorial part of the data on one model. For that part, corollary 5.38.6 gives a canonical ordinary type by combining a Gauduchon bound for the mass of the divisorial current with a smooth decomposition of currents; it does not determine the trace data of higher codimension, which appear on further modifications. The exceptional divisor of theorem 20.4.26 describes the discrepancy between the canonical types on two models, together with a cocycle relation and approximation in linear mass. By example 20.4.31, a nonzero exceptional term is compatible with an exact trace potential: the term records information absent from the divisorial part on the original model. Enlarging the background provides a competitor with controlled curvature, but it does not automatically exclude additional divisorial orders. In the proof for curve fibrations, every order on a further model reduces to a multiplicity in one variable. Under the two mass properties, theorem 13.21.2 controls Lelong numbers through the deficit of upper volume and a positive lower volume. The missing step is to extend such control beyond these mass conditions.

Even exact realization does not control the mass of all representatives of the ideal class. In proposition 20.3.62, on a product with the Hopf threefold, the realizing potential has finite extended mass, while another potential with the same multiplier ideals and the same trace numbers has infinite mass. A proposed characterization by volumes or by continuity must specify which representative, or which class of masses, it uses.

Theorem 20.3.48 answers the realization question for the Hopf pair $(H_{n+1},H_ n)$, $n\geq 3$, when the ambient potential is pulled back from $\mathbb P^ n$ and has zero generic Lelong number along the projective hyperplane. Every Hermitian metric on $H_ n$ fails bounded mass, yet the same pulled-back potential realizes the trace data on all compact models. Within the invariant backgrounds $a\alpha +b\beta $, corollary 20.3.55, proposition 20.3.57 give an ideal class compatible with changes of background, addition, positive scaling and decreasing analytic approximation. Exactness persists in every larger Hermitian background. This settles the projective pullback case; general ambient data and background compatibility outside this family remain part of the problem.

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.