Wall-chamber decompositions for generalised Monge-Amp\`ere equations

Sohaib Khalid; Zakarias Sj\"ostr\"om Dyrefelt

REVIEW 2 major objections 5 minor 2 cited by

Wall-chamber decompositions for generalised Monge-Amp\`ere equations

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Pith's one-line read A finite list of subvarieties decides when generalized Monge-Ampère equations are solvable.

desk verdict Real finiteness theorem for destabilizing subvarieties in higher dimensions, but the optimal-destabilizer extension has a repairable gap for curves. read the letter →

arxiv 2412.20089 v2 pith:ATFEMDXU submitted 2024-12-28 math.DG math.AGmath.AP

Sohaib Khalid , Zakarias Sjöström Dyrefelt This is my paper · ORCID

classification math.DGmath.AGmath.AP MSC 14J6032Q2653C0753C5553E30

keywords generalizedMonge-AmpèreequationsJ-equationNakai-MoishezoncriterionmodifiedKählerconeswall-chamberdecompositionsdestabilizingsubvarietiesdeformedHermitianYang-MillsequationinverseHessian

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to turn the numerical solvability criteria for generalized Monge-Ampère equations from an infinite check into a finite one. For the J-equation and its relatives, solvability is known to be equivalent to slope inequalities tested on every subvariety; here the authors argue that under a mild positivity condition on the classes $\tau_p = \mu\alpha - p\beta$, only finitely many subvarieties can violate these inequalities, and those are rigid. The same mechanism is extended to inverse Hessian equations, some deformed Hermitian Yang-Mills equations, and some Z-critical equations through the notion of factor classes. If the argument is right, the nonlinear Nakai-Moishezon criterion collapses to a finite list, and the space of Kähler classes decomposes into locally finite wall-chamber regions on which solvability is constant.

What carries the argument

The engine is the non-Kähler locus $E_{nK}(\tau)$ of a big $(1,1)$-class and the modified Kähler cone, meaning classes whose non-Kähler locus contains no subvarieties of dimension at least $p+1$. The key positivity lemma says that if a $p$-dimensional subvariety $V$ has non-positive intersection with a big class $\tau$, then $V$ lies in $E_{nK}(\tau)$. Hence, when $\tau_p$ is $(p+1)$-modified Kähler, every destabilizing $p$-dimensional subvariety must be one of the finitely many $p$-dimensional components of $E_{nK}(\tau_p)$. For generalized Monge-Ampère equations, factor classes $\tau_p$ are extracted from the polynomial $Q_p(x,y) = (\exp(x)(1-P(y)))[p]$, and these carry the same finiteness argument.

What would settle it

Look for a compact Kähler manifold and Kähler classes $\alpha,\beta$ satisfying the $(p+1)$-modified Kähler condition for every $p$, yet having infinitely many irreducible subvarieties $V$ with $\mu(V) \ge \mu(X)$. A direct intersection-theoretic computation on any such example would refute Theorem 3.9; conversely, the paper's own blow-up example, where $\mu\alpha - (n-1)\beta$ is not big, already shows the hypothesis cannot simply be dropped.

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Extended reading notes

Core claim

The central claim is Theorem 3.9: if for every $p = 1,\dots,n-1$ the class $\tau_p = \mu\alpha - p\beta$ is $(p+1)$-modified Kähler, then the set of irreducible subvarieties $V$ with $\mu(V) \ge \mu(X)$ is finite. Combined with the known equivalence between solvability and J-stability, this yields Theorem 1.6: the J-equation is solvable exactly when a finite list $V_1,\dots,V_r$ satisfies the strict slope inequalities $\mu > \mu(V_i)$. The same mechanism, once a generalized Monge-Ampère equation is shown to be factorizable into classes $\tau_p$, gives finite effective testing for these equations and locally finite wall-chamber decompositions.

Load-bearing premise

The load-bearing assumption is that each class $\tau_p = \mu\alpha - p\beta$ (or the corresponding factor class) is $(p+1)$-modified Kähler, meaning its non-Kähler locus contains no subvarieties of dimension $p+1$ or higher; if this fails, infinitely many destabilizing subvarieties can occur.

Editorial extensions

If this is right

The J-equation can be decided by checking strict slope inequalities on a finite list of subvarieties, not on all subvarieties.
The boundary of the J-stable locus is a locally finite union of codimension-one walls, each cut out by one destabilizing subvariety.
The same holds for generalized Monge-Ampère equations, including inverse Hessian equations, under the factor-class positivity assumption.
On threefolds, destabilizing subvarieties are rigid: surfaces are unique effective cycles in their homology class, and curves have negative self-intersection after resolution.
Explicit families of compact Kähler manifolds in every dimension satisfy the hypotheses, giving the first higher-dimensional PDE wall-chamber decompositions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves open whether the modified Kähler hypothesis can be weakened; the blow-up example shows some positivity is necessary, but the optimal condition is not identified.
A consequence the authors only hint at is that the finite list could be turned into an algorithm: compute the finitely many components of the non-Kähler loci and test only those subvarieties.
If the wall-chamber picture is correct, one should expect the J-flow's bubbling locus to coincide with the finite destabilizing set; checking this on the explicit projective-bundle examples would be a natural testable extension.
The factorizability of all generalized Monge-Ampère equations suggests that the finiteness phenomenon is ultimately a property of the polynomial data defining the equation, not of the individual PDE.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

Two things to know up front. Theorem 3.9 is the backbone: under the assumption that each class μα − pβ is (p+1)-modified Kähler, there are only finitely many destabilizing subvarieties, and the proof is clean. That is a genuine advance over the surface case and over prior criteria. The second thing is that the extension to optimal destabilizers, Theorem 3.10, is not fully proved as written. The proof passes to β′ = β − Δβ(α)α, making the triple semistable, and then invokes Lemma 2.4 for p = 2,…,n−1. For p = 1, there is no modified Kähler hypothesis, and the argument does not rule out infinitely many curves C with μ(C) = μ. The stress-test note describes this exactly; I checked the proof and it simply asserts finiteness after the p≥2 discussion. This gap affects Theorem 1.1(4), Corollary 4.5, and Corollary 4.7.

The paper does a lot well. The factor-class formalism for gMA equations is useful, and the factorization lemma (5.6) is a nice observation. The concrete examples—blowups, Wu’s projective bundles, threefold computations—show the positivity hypotheses are not empty. Theorem 3.1 for threefolds is fully handled, including the subtle curve case, and the wall-chamber theorem 6.2 follows cleanly once the finiteness results are in place. The citations to the authors’ own earlier work are appropriate; the surface result [22] is genuinely prior.

Soft spots beyond Theorem 3.10: Lemma 4.4’s proof is a sketch, and the step where L(s) = R(s) is asserted does not follow from the preceding sentences. I suspect the intended linearity argument can be made to work, but as written it is incomplete. The modified Kähler assumptions are strong; Example 4.1 shows some hypothesis is needed, but the paper does not give a general criterion beyond the projection trick of Section 4.3. That is a limitation, not a flaw.

Who should read this: anyone working on J-equation, inverse Hessian equations, dHYM, or the PDE/stability analogy. It deserves a serious referee. My recommendation is to send it out, with the request that the authors fix the p=1 case in Theorem 3.10 and expand the proof of Lemma 4.4. The core finiteness theorem is solid, and the rest is likely repairable.

Referee Report

2 major / 5 minor

Summary. The paper studies generalized Monge-Ampère (gMA) equations on compact Kähler manifolds, focusing on the algebraic structure of subvarieties that violate the Nakai–Moishezon type numerical criteria for solvability. The main results assert that, under modified-Kähler positivity assumptions, the set of destabilizing subvarieties is finite, that optimal destabilizers are finite, that such subvarieties are rigid in a suitable sense, and that these finiteness statements imply locally finite wall-chamber decompositions for the J-equation and certain gMA/dHYM equations. The main theorems are Theorem 3.1 (threefolds), Theorem 3.9 (arbitrary dimension under (p+1)-modified Kähler assumptions), and Theorem 3.10 (finiteness of optimal destabilizers). Applications include effective solvability criteria, examples on projective bundles and blow-ups, and wall-chamber decompositions.

Significance. If the main results are correct, this is a substantial advance: it reduces the infinite Nakai–Moishezon type test for a large class of geometric PDE to a finite list of subvarieties, and it provides the first higher-dimensional examples of PDE analogues of Bridgeland's locally finite wall-chamber decompositions. The proof strategy is conceptually clean, reducing the problem to the non-Kähler locus of certain cohomology classes and using finiteness of irreducible components of analytic sets. The paper also contains explicit and checkable examples, especially the Wu-type projective bundle constructions, and the proof of Theorem 3.9 is short and correct as written. The main advertised applications, however, rely on Theorem 3.10 and Lemma 4.4, both of which have proof gaps that are load-bearing.

major comments (2)

[§3.3, Theorem 3.10] The proof of Theorem 3.10 does not handle the case p=1, which is required for finiteness of optimally destabilizing curves. The proof passes to β' = β − Δ_{β}(α)α, so that (X,α,β') is semistable and Δ_{β'}(α)=0. For p≥2 the argument via Lemma 2.4 works, but for p=1 the hypothesis (6) is not imposed on τ₁ = μ α − β'. A curve C with equality μ(C)=μ satisfies ∫_C(μ α − β')=0, hence Lemma 2.4 only shows C ⊆ EnK(τ₁). Since τ₁ = τ₂ + β' with τ₂ ∈ M₃K and β' Kähler, τ₁ is in M₃K but not necessarily in M₂K; EnK(τ₁) may therefore contain surface components, and nothing in the proof shows that (μ α − β')|_S is big on such a surface S, nor that only finitely many curves in S achieve equality. Thus the claims of finiteness of Destopt in Theorem 1.1(4), Corollary 4.5, and Corollary 4.7 are not established. The non-emptiness assertion, attributed to [28], also appears to assume the finiteness it is meant to prove; a precise citation or proof is needed.
[§4.3, Lemma 4.4] The proof of Lemma 4.4 is incomplete as written. The text reaches "Putting it all together we find that L(s) = R(s) for all s ≥ 0" and then "Thus, we have the inequality", but no derivation of L(s)=R(s) is given, and the displayed inequality is missing. The logical steps L(0)=R(0), the implication L(s)≥0 ⇒ R(s)≥0, and the existence of some s with L(s)≤0 do not by themselves imply identity of the two affine functions. The first inequality in (9) is asserted to follow "similarly" but is not proved. Since Lemma 4.4 is the key input for Proposition 4.2 and hence for the theorems producing finite optimal destabilizers (Corollaries 4.5 and 4.7, Theorem 4.6), this gap must be repaired with a complete argument.

minor comments (5)

[§3.3, Theorem 3.9 proof] In the proof of Theorem 3.9, the inclusion is written as Z ⊆ EnK(µα,βα − pα), but the class should be µα,βα − pβ; this appears to be a typo.
[§4.3, Proposition 4.2] The notation θ ≥ θ′ for θ − θ′ nef is used in the computation following equation (10) but is not defined before its first use; please define it explicitly.
[§4.3, Lemma 4.4 proof] The definition of the second path β_s := (1−s)α_{t_s} + sη* with t_s chosen so that α_{t_s}=α_t for all s ∈ [0,1) is hard to follow; the dependence of t_s on s should be spelled out, and the assertion that L(s) is linear in the relevant argument should be justified by a precise reference to [28] or by an argument.
[Abstract and references] The abstract contains a typo, "familes"; also, reference [2] has an unusual combined entry (thesis and journal pages) that should be cleaned up.
[§5.2, Theorem 5.15] In the proof of Theorem 5.15, the claim that the classes α − cot((φ+πl)/p)β are Kähler for l=1,...,p−1 under the stated hypotheses is correct, but the verification for p=2 and p=3 is compressed; a short sentence listing the ranges of the cotangents would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the finiteness theorems are conditional on explicit modified-Kähler hypotheses and reduce to non-Kähler loci via Lemma 2.4; the cited prior work is independent published mathematics.

full rationale

The paper's central claims are conditional statements: if the classes τ_p = μ α − pβ (or their shifted versions) are (p+1)-modified Kähler, then the set of p-dimensional destabilizing subvarieties is finite. The reduction is direct: μ(V) ≥ μ is rewritten as ∫_V (μ α − pβ)·α^{p−1} ≤ 0, and Lemma 2.4 (proved from Boucksom's and Demailly's results) forces V ⊆ EnK(μ α − pβ). Because τ_p ∈ M_{p+1}K, Lemma 2.3 says EnK(τ_p) has no component of dimension ≥ p+1, so the p-dimensional subvarieties contained in it are among finitely many irreducible components. No parameter is fitted to the data being predicted, and the target statement is not used as an input; the modified-Kähler condition is an assumption, not a consequence of the conclusion. The paper does cite the authors' earlier work: [22] for the surface case and [28] for the stability threshold and its linearity. These are published, independently proved results that do not assume the present theorems, so under the review rules they are real evidence rather than circularity. The factor classes τ_p for generalized Monge-Ampère equations are obtained from an explicit polynomial factorization (Lemma 5.6), not by defining the answer into the input. One non-circular concern should be flagged: the proof of Theorem 3.10 explicitly does not impose the modified-Kähler condition for p=1, and the argument 'this combined implies Destopt_{α,β′} is finite' does not handle curves contained in surface components of EnK(μ α − β′); this is a proof gap or correctness issue, not a circularity, and it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new entities or free parameters; the factor classes r_p are determined by the coefficients of the equation. The main axioms are standard Kähler geometry results and prior numerical criteria, plus the positivity hypotheses on the classes μ α - pβ.

assumptions (6)

domain assumption Compact Kähler manifold X with Kähler classes α,β; the numerical criteria of [5], [12], [19] characterize solvability by testing all proper subvarieties.
Used throughout, e.g., Theorem 5.1 (Datar-Pingali, Fang-Ma) and the J-equation criterion from Gao Chen.
standard math For a big class τ, the non-Kähler locus EnK(τ) is a proper analytic subset, and if V is not contained in it then ∫_V τ·ω1·...·ω_{p-1}>0 (Lemma 2.4).
Proved in Section 2 using Demailly regularization and Boucksom's results.
domain assumption On a smooth projective surface, for a big class L, only finitely many irreducible curves C satisfy L·C≤0 (from [22, Prop 3.1, Thm 5.3]).
Used in the threefold proof (Theorem 3.1) and gMA/dHYM analogues to bound destabilizing curves inside surfaces.
domain assumption The stability threshold Δ is linear in β and its infimum is realized by finitely many subvarieties ([28]).
Used in Theorem 3.10 and Lemma 4.4 to shift β to a semistable class; the proof of Lemma 4.4 is incomplete as written.
standard math Hironaka resolution of singularities exists for compact Kähler surfaces and preserves the relevant intersection numbers (projection formula).
Used in the singular surface case of Theorem 3.1 and in the rigidity statements.
standard math The p-modified Kähler cones are open, and Lemma 2.3 characterizes them via dimensions of components of EnK.
Proved in Section 2; used for the openness argument in Lemma 2.5 and for wall-chamber local finiteness.

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Cite this review

Pith. "Pith review of Wall-chamber decompositions for generalised Monge-Amp\`ere equations." pith.science (2026). https://pith.science/paper/ATFEMDXU

@misc{pith2026241220089,
  author       = {Pith},
  title        = {Pith review of: Wall-chamber decompositions for generalised Monge-Amp\`ere equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ATFEMDXU}},
  note         = {Machine review of arXiv:2412.20089}
}

read the original abstract

Generalised Monge-Amp\`ere equations form a large class of PDE including Donaldson's J-equation, inverse Hessian equations, some supercritical deformed Hermitian-Yang Mills equations, and some Z-critical equations. Solvability of these equations is characterised by numerical criteria involving intersection numbers over all subvarieties, and in this paper, we aim to characterise algebraically what happens when these nonlinear Nakai-Moishezon type criteria fail. As a main result, we show that under mild positivity assumptions, there is a finite number of subvarieties violating the Nakai type criterion, and such subvarieties are moreover rigid in a suitable sense. This gives first effective solvability criteria for these families of PDE, thus improving on work of Gao Chen, Datar-Pingali, Song and Fang-Ma, and provides first existence results in higher dimension of compact K\"ahler manifolds exhibiting a natural PDE analog of Bridgeland's locally finite wall-chamber decomposition.

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Forward citations

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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Khalid, Z

S. Khalid, Z. Sjöström Dyrefelt, The set of destabilizi ng curves for deformed Hermitian Yang- Mills and Z-critical equations on surfaces, Int. Math. Res. Not., Volume 2024, Issue 7, April 2024, Pages 5773–5814

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Calabi, The space of Kähler metrics, Proceedings of th e International Congress of Mathematics 1954 2 (1954), 206–

E. Calabi, The space of Kähler metrics, Proceedings of th e International Congress of Mathematics 1954 2 (1954), 206–

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