Capacity Stability of Complex Monge-Ampère Equations with Moving Prescribed Singularities

Abstract: For complex Monge-Ampère equations with moving big cohomology classes and prescribed model singularities of positive Monge-Ampère mass, we prove that, under total variation convergence of the right-hand side non-pluripolar positive Radon measures, convergence of the prescribed model potentials in Monge-Ampère capacity is equivalent to convergence in capacity of the associated normalized solutions. We further prove that the ceiling operator coincides with the singularity envelope for potentials associated to a big $(1,1)$-class, regardless of their Monge-Ampère mass, thereby resolving a conjecture of Darvas-Di Nezza-Lu. Consequently, the singularity envelope is idempotent without the positivity assumption on the mass.

• The paper proves that solutions of complex Monge–Ampère equations converge in capacity if and only if their moving prescribed model singularities do, assuming matching masses and total variation convergence of the measures.
• The authors develop strict subbarriers, uniform relative-capacity domination, and volume-recovery estimates to control moving backgrounds and extend stability beyond the more rigid d_S topology.
• The paper also proves the ceiling operator equals the singularity envelope and that the envelope is idempotent without positive-mass assumptions, with applications to weak geodesics, twisted Kähler–Einstein equations, and moving quantization.

This paper, by Pang, Sun, Wang and Zhou (2607.12797), establishes that the Monge–Ampère capacity is the intrinsic topology governing stability of complex Monge–Ampère equations with prescribed singularities when the singularity data move. Along the way it resolves two open conjectures of Darvas–Di Nezza–Lu concerning the ceiling operator and the idempotence of the singularity envelope in arbitrary mass.

Setting and motivation

Let $(X,\omega)$ be a compact Kähler manifold and $\theta$ a smooth closed real $(1,1)$-form with big cohomology class. The relative pluripotential theory of Darvas–Di Nezza–Lu provides existence, uniqueness, comparison principles, and variational tools for equations

$$\theta_u^n = \mu,\qquad u\in\mathcal E(X,\theta,\phi),$$

where $\phi$ is a positive-mass model potential ($\phi=P_\theta[\phi]$, $\int_X\theta_\phi^n>0$) prescribing the singularity type, and $\mu$ is a non-pluripolar measure of total mass $\int_X\theta_\phi^n$. Darvas–Di Nezza–Lu proved a stability theorem when the prescribed singularity types converge in the $d_{\mathcal S}$-metric on singularity types. However, $\theta$0 records fine singularity data (Lelong numbers, multiplier ideals) and is too rigid for geometric families: logarithmic poles may move or collide so that potentials converge in capacity while their singularity types do not converge in $\theta$1. Since capacity convergence is the natural stability notion for the complex Monge–Ampère operator, the paper asks which intrinsic topology on moving prescribed singularities is detected by Monge–Ampère stability. The answer is capacity itself.

Capacity stability for moving singularities

The first main theorem assumes $\theta$2 uniformly with all classes big, $\theta$3 normalized positive-mass model potentials converging in $\theta$4, and non-pluripolar measures $\theta$5 in total variation with matching masses. If $\theta$6 are the normalized solutions of $\theta$7, $\theta$8, then

$\theta$9

The forward implication refines the $(1,1)$0-stability theorem of Darvas–Di Nezza–Lu; the reverse implication shows capacity convergence of solutions forces capacity convergence of the prescribed model potentials under the no-mass-loss assumption, so the capacity topology is not merely convenient but exactly optimal.

The proof of the forward direction introduces two technical devices of independent interest. First, a strict subbarrier lemma: if $(1,1)$1 has positive non-pluripolar mass, there exist $(1,1)$2 and constants $(1,1)$3 with $(1,1)$4 and $(1,1)$5. This replaces unavailable $(1,1)$6-control by a quantitative slope gap; combined with minimal truncations, it pushes comparison errors into deep singularity tails, which are small by capacity convergence of the model potentials. Second, a uniform domination estimate generalizing the relative capacity bound of DDL: after discarding finitely many indices,

$(1,1)$7

with $(1,1)$8 independent of $(1,1)$9, proved via integration-by-parts formulas for non-pluripolar products and mixed-mass monotonicity. The argument then proceeds through an eight-step scheme: $$\theta_u^n = \mu,\qquad u\in\mathcal E(X,\theta,\phi),$$0-convergence via compactness and uniqueness, capacity convergence of the moving minimal potentials $$\theta_u^n = \mu,\qquad u\in\mathcal E(X,\theta,\phi),$$1, tail estimates from total variation convergence, a fixed barrier construction, a core capacity estimate on good sets, parameter selection, and Hartogs' lemma for the upper tail.

A concrete example on $$\theta_u^n = \mu,\qquad u\in\mathcal E(X,\theta,\phi),$$2 demonstrates strictness. With Fubini–Study metric and moving divisors $$\theta_u^n = \mu,\qquad u\in\mathcal E(X,\theta,\phi),$$3, set $$\theta_u^n = \mu,\qquad u\in\mathcal E(X,\theta,\phi),$$4 for fixed rational $$\theta_u^n = \mu,\qquad u\in\mathcal E(X,\theta,\phi),$$5 and $$\theta_u^n = \mu,\qquad u\in\mathcal E(X,\theta,\phi),$$6. Then $$\theta_u^n = \mu,\qquad u\in\mathcal E(X,\theta,\phi),$$7 in capacity and the non-pluripolar masses are constant, equal to $$\theta_u^n = \mu,\qquad u\in\mathcal E(X,\theta,\phi),$$8; yet for $$\theta_u^n = \mu,\qquad u\in\mathcal E(X,\theta,\phi),$$9 the rooftop $\phi$0 has minimal singularity type, so the $\phi$1 quantity satisfies

$\phi$2

giving a uniform positive lower bound on $\phi$3. Thus the capacity-stability theorem applies to families genuinely outside the scope of the $\phi$4-stability results of Darvas–Di Nezza–Lu and Do–Vu.

Continuity of envelopes and optimality

The reverse implication rests on a capacity continuity theorem for relative envelopes: if $\phi$5 in capacity among normalized positive-mass model potentials, $\phi$6 quasi-continuous with $\phi$7 in capacity, and $\phi$8, $\phi$9 q.e., then $\phi=P_\theta[\phi]$0 in capacity. The proof combines the envelope contact theorem ($\phi=P_\theta[\phi]$1 concentrated on $\phi=P_\theta[\phi]$2), the slope-gap subbarrier, and a lemma showing that q.e. upper bounds pass to $\phi=P_\theta[\phi]$3 limits against non-pluripolar test measures. Rooftop envelopes inherit this continuity, which is the key input for recovering singularity envelopes from solutions.

For optimality, the paper proves a no-mass-loss recovery theorem: if normalized solutions $\phi=P_\theta[\phi]$4 in capacity with $\phi=P_\theta[\phi]$5 in total variation, then the singularity envelopes $\phi=P_\theta[\phi]$6 in capacity. The mechanism is a volume recovery estimate: solving auxiliary equations $\phi=P_\theta[\phi]$7 in $\phi=P_\theta[\phi]$8, using the uniform relative $\phi=P_\theta[\phi]$9 estimate to obtain $\int_X\theta_\phi^n>0$0, and applying the rooftop contact inequality to bound $\int_X\theta_\phi^n>0$1 by relative tails $\int_X\theta_\phi^n>0$2, which are small uniformly in $\int_X\theta_\phi^n>0$3 by total variation convergence. Fixed-level rooftops $\int_X\theta_\phi^n>0$4 converge by the envelope continuity theorem, and passing $\int_X\theta_\phi^n>0$5 identifies the limit as the singularity envelope. Combined with the identity $\int_X\theta_\phi^n>0$6 (valid because $\int_X\theta_\phi^n>0$7), this yields the equivalence asserted above.

The ceiling conjecture and idempotence

The second main result resolves two conjectures. For any $\int_X\theta_\phi^n>0$8, where $\int_X\theta_\phi^n>0$9 is big but no positivity of mass is assumed:

$\mu$0

Here $\mu$1 is the ceiling operator of Darvas–Di Nezza–Lu, defined via preservation of all mixed masses $\mu$2; they had established $\mu$3 only for positive mass and conjectured the general case, while Lu conjectured idempotence of the singularity envelope. The proof approximates $\mu$4 by $\mu$5, whose mixed masses equal those of $\mu$6 with $\mu$7; since $\mu$8, each $\mu$9 is a positive-mass model potential and $\int_X\theta_\phi^n$0. A normalized tail estimate—using multilinearity of the non-pluripolar product to decompose $\int_X\theta_\phi^n$1 into mixed products $\int_X\theta_\phi^n$2—plus a volume recovery argument parallel to the one above gives $\int_X\theta_\phi^n$3 in $\int_X\theta_\phi^n$4, hence $\int_X\theta_\phi^n$5. As a corollary, the full-mass characterization of DDL extends to zero mass: for $\int_X\theta_\phi^n$6, equality of all mixed masses is equivalent to $\int_X\theta_\phi^n$7.

Applications

Weak geodesic segments. Using a rooftop–Legendre representation derived from Kiselman's minimum principle and one-dimensional Fenchel–Moreau duality—with the supremum reducible to a compact interval $\int_X\theta_\phi^n$8—the paper proves that weak geodesic segments with endpoints bounded between $\int_X\theta_\phi^n$9 and $d_{\mathcal S}$0 converge uniformly in capacity whenever the prescribed singularities and endpoints do. This is a capacity-topology analogue of $d_{\mathcal S}$1-stability of geodesics.

Twisted Kähler–Einstein equations in big classes. Building on the existence theorem of Darvas–Zhang for $d_{\mathcal S}$2, the paper proves compactness: if tame measures $d_{\mathcal S}$3 have twists converging appropriately ($d_{\mathcal S}$4 uniformly, $d_{\mathcal S}$5 in capacity, $d_{\mathcal S}$6), then every subsequence of normalized solutions admits a further subsequence converging in capacity to a solution of the limiting twisted equation. The proof chains uniform Ding properness (from domination $d_{\mathcal S}$7), a uniform Skoda-type integrability estimate on Ding-bounded sets obtained via Guan–Zhou's strong openness theorem and Guan–Li–Zhou's stability of multiplier ideal sheaves, Hölder interpolation to get uniform $d_{\mathcal S}$8 bounds on the right-hand side densities, and finally the relative $d_{\mathcal S}$9 estimate together with the conditional stability theorem.

Moving quantization of partial equilibrium measures. For partial $\theta$00-equilibrium envelopes $\theta$01 in the sense of Darvas–Xia, capacity convergence of the $\theta$02-model projections $\theta$03 and uniform convergence of weights imply $\theta$04 in capacity; with mass convergence this yields weak convergence of the equilibrium measures. Combined with the Darvas–Xia quantization theorem, one obtains the iterated limit

$\theta$05

On $\theta$06 with moving poles, the multiplier ideals compute explicitly as $\theta$07, so degree-$\theta$08 sections vanishing to order $\theta$09 along the moving divisor quantize the partial equilibrium measure—a moving-singularity analogue of Bergman measure convergence in the spirit of Fekete point equidistribution.

Limitations and open questions

Several hypotheses carry real weight. The forward stability theorem requires total variation convergence of the right-hand sides and positive mass of the limit model potential; the reverse implication additionally requires that the solutions lie in the corresponding relative full-mass classes and that the Monge–Ampère measures converge in total variation—the no-mass-loss condition is essential to recover the envelopes. The twisted Kähler–Einstein application assumes $\theta$10, uniform lower bounds on the twisting potentials, and tameness of the measures; without these the uniform Ding properness and Skoda estimates fail. In the quantization section, the authors state explicitly that the double limit $\theta$11 is not known to be exchangeable; they conjecture that under uniform Bernstein–Markov conditions on the sampling measures the diagonal limit holds, which would yield more flexible moving-singularity quantizations relevant to Fekete point equidistribution. Whether quantitative rates accompany the qualitative capacity equivalences established here remains unaddressed.

Conclusion

The paper identifies Monge–Ampère capacity as the optimal topology for stability of complex Monge–Ampère equations with moving prescribed singularities in big cohomology classes, strictly enlarging the reach of the earlier $\theta$12-stability theory, and settles the Darvas–Di Nezza–Lu ceiling conjecture and Lu's idempotence conjecture without any positivity assumption on the mass. The supporting machinery—strict subbarriers with slope gaps, uniform relative capacity domination over moving backgrounds, and volume recovery from rooftop contact inequalities—is likely to serve in further stability questions for degenerate fully nonlinear equations on compact Kähler manifolds.