The Critical LYZ Equation in K\"ahler Geometry

Dekai Zhang; Jixiang Fu; Shing-Tung Yau

REVIEW 3 major objections 4 minor 2 cited by

The Critical LYZ Equation in K\"ahler Geometry

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Pith's one-line read Proven: smooth solution for critical LYZ equation under a subsolution

desk verdict The paper attacks a real open problem and much of the proof is sound, but the new Liouville theorem has an unjustified scaling normalization that leaves the main theorem unsupported. read the letter →

arxiv 2511.21492 v4 pith:BAU6TUXD submitted 2025-11-26 math.DG math.APmath.CV

Jixiang Fu , Shing-Tung Yau , Dekai Zhang This is my paper · ORCID

classification math.DGmath.APmath.CV MSC 32W2053C55

keywords LYZequationdeformedHermitianYang-MillscriticalphasesubsolutioncomplexHessianestimatesgradientLiouvilletheoremquotientequations

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

The paper establishes existence and uniqueness of smooth solutions to the LYZ equation at its critical phase θ=(n−2)π/2 on any compact Kähler manifold, provided a subsolution satisfying a pointwise angular condition exists. This is the endpoint case of a family of equations previously solved only above the critical phase. The proof reaches the endpoint by solving nearby supercritical equations and showing their solutions stay uniformly bounded in C^{2,α} independently of how close the phase is to critical. A new Liouville theorem for the limiting equation σ_{n−1}+σ_n=0 on C^n supplies the missing gradient control. Success at the critical phase also yields the 3D Hessian equation σ_2=1 and the 4D Hessian quotient σ_3=σ_1 under weaker hypotheses than were previously available.

What carries the argument

Central object: the LYZ (deformed Hermitian Yang-Mills) equation θ_ω(χ_u)=θ, with θ_ω(χ_u)=Σ_{i=1}^n arctan λ_i and λ_i the eigenvalues of χ+√−1∂∂̄u with respect to ω. At the critical phase θ=(n−2)π/2 the target equation is approached by the supercritical family (6), and the proof's engine is a uniform complex Hessian estimate sup |√−1∂∂̄u_t|_ω ≤ C(1+sup|∇u_t|²), independent of t and of (θ(t)−(n−2)π/2)^{-1}. The gradient estimate then comes from rescaling a hypothetical blowing-up sequence; the limit v is a bounded nonconstant weak solution of σ_n=0, σ_{n−1}=0, or σ_{n−1}+σ_n=0. The new case is tamed by the lifting identity: v solves σ_{n−1}+σ_n=0 on C^n if and only if v+|z_{n+1}|² solves σ_

What would settle it

Check the scaling normalization (52) directly: for any bounded C^1 function v with oscillation A and sup|∇v|=1, the rescaled function ṽ(z)=A^{-1}(v(√A z)−inf v) has oscillation 1 but gradient bound A^{-1/2}, which exceeds 1 when A<1. If such a v satisfies all other hypotheses of Theorem 4.1 (nonconstant, (n−1)-subharmonic, v+|z|² plurisubharmonic, weak solution and viscosity supersolution of σ_{n−1}+σ_n=0, with ∆v≤C weakly), then the Liouville theorem is false and the existence proof breaks at that point.

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Extended reading notes

Core claim

On a compact Kähler manifold (M,ω), fix a closed real (1,1)-form χ whose total phase ∫_M (χ+√−1ω)^n has principal argument π. The paper's main theorem says that if some smooth u satisfies the critical subsolution condition min_j Σ_{i≠j} arctan λ_i(χ_u) > (n−3)π/2, then the critical LYZ equation θ_ω(χ_u)=(n−2)π/2 has a unique smooth solution normalized by sup u=0. The theorem reaches this endpoint by perturbing to supercritical phase θ(t)=(n−2)π/2+O(t), where solutions were already known to exist, and then proving C^{2,α} bounds on this family that stay uniform as t→0. The two new ingredients are a second-order bound sup |√−1∂∂̄u_t|_ω ≤ C(1+sup|∇u_t|²) that does not blow up at the critical ph

Load-bearing premise

The load-bearing step is the normalization in (52): after rescaling, the paper assumes the gradient of v is no larger than the square root of v's total oscillation, and this inequality is not a consequence of the C^1 bound proved earlier; if it cannot be arranged, the Liouville theorem and with it the existence proof collapse.

Editorial extensions

If this is right

Under the stated subsolution condition, the critical LYZ equation has a unique smooth solution normalized by sup u=0.
The solutions of the approximating supercritical family converge in C^{2,α} to the critical solution, since the estimates do not depend on the distance to the critical phase.
In dimension 3, the critical equation is the Hessian equation σ_2(χ_u)=1, so the theorem solves it under the subsolution-type condition χ∧ω>0 plus integral normalizations, without requiring χ∈Γ_2(M).
In dimension 4, the critical equation is the Hessian quotient σ_3(χ_u)=σ_1(χ_u), solved under 3χ²∧ω−ω³>0 plus integral normalizations, weaker than χ∈Γ_3(M).
Solvability at the critical phase is a concrete step toward realizing the LYZ equation as a stability condition on the bounded derived category.

Reading between the lines

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

The normalization (52) in the Liouville proof is the load-bearing step; before relying on it, a reader should verify the rescaling claim, since the stated C^1 bound alone does not produce |∇v|≤(osc)^{1/2}.
The lifting identity suggests that other techniques from the homogeneous complex Monge-Ampère equation may transfer to the critical LYZ equation; for example, one might look for interior Hessian estimates for σ_{n−1}+σ_n=0 by working in one dimension higher.
The subcritical regime θ<(n−2)π/2 is likely to behave differently: known examples on R^n show C^{1,α} and Lipschitz solutions that are not smooth, so the compact-manifold subcritical case may genuinely admit singular solutions, making this endpoint result the sharp smooth boundary.
The applications reveal that the integral normalizations in dimensions 3 and 4 are necessary; this hints that the subsolution condition (5) may itself be necessary for existence in the critical case, though the paper does not prove the converse.

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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

Short version: this is a serious paper on an open problem that matters, and most of the machinery is sound, but the new Liouville theorem has a real gap. I wouldn't accept it yet; I would send it to a referee.

What's genuinely new: the paper solves the critical LYZ phase θ=(n−2)π/2 under the subsolution condition, which Collins–Jacob–Yau and Li explicitly left open. The continuity method through supercritical equations is standard, but the paper gets the phase control and the uniform estimates right. The lifting trick v+|z_{n+1}|^2 turns σ_{n-1}+σ_n=0 into σ_n=0 on C^{n+1}; that's a nice idea and likely reusable. The 3D σ2=1 and 4D σ3=σ1 corollaries are credible, and the necessity arguments in Section 5 look correct. The complex Hessian estimate (Theorem 3.1) is long but coherent, and the isolation of the new Liouville theorem is honest.

The soft spot is Theorem 4.1. The proof's normalization (52) is not justified. For v with ||v||_{C^1}≤C, setting C1=(sup v−inf v)^{1/2} and \tilde v(z)=C1^{-2}(v(C1z)−inf) gives |∇\tilde v|≤C/C1, not ≤C1. The claimed bound needs (sup−inf)≥C^2, which is not in the assumptions. That matters because Case 2's comparison argument uses the normalized bounds 0≤v≤1, |∇v|≤1 to control everything from the constants 1/60, 9/10, 3/10 to the Dinew–Kołodziej/Székelyhidi subharmonicity step. Without a way to get those bounds, the Liouville theorem is unproved. And Theorem 4.1 is exactly what excludes the blow-up limit in Subcase 2.2 of the gradient estimate, so the chain from Theorem 4.1 to Theorem 1.1 is broken as written. There are also places where the proof says 'following [38] and [10]' without giving enough detail, which makes it hard to tell whether this is a small fix or a deeper issue.

Is it fatal? Not obviously. The result may be true, and the rest of the architecture is plausible. But the central existence theorem currently rests on an unsupported lemma. A referee who knows [10,38] and the Liouville literature should be asked whether the normalization can be repaired—maybe by a different scaling, or by deriving sup−inf≥C^2 from the equation. If that can be fixed, this is a significant paper.

Audience: people working on the deformed Hermitian–Yang–Mills equation, Hessian equations, and Bridgeland stability. It deserves peer review, not desk rejection, and the referee should be asked to focus on Theorem 4.1 first.

Referee Report

3 major / 4 minor

Summary. The paper claims to resolve the critical case of the LYZ/dHYM equation, i.e. the existence of smooth solutions to θ_ω(χ_u) = (n−2)π/2 on a compact Kähler manifold under the subsolution condition (5). The strategy is to perturb the critical phase by a small parameter t, obtain supercritical solutions u_t via Lin's theorem, and then prove uniform C^{2,α} estimates for u_t that are independent of t and of the phase gap. The main technical novelties are a Hou–Ma–Wu type Hessian estimate (Theorem 3.1) with uniform constants, and a new Liouville theorem for σ_{n−1}+σ_n=0 on C^n (Theorem 4.1), proved by lifting v to v+|z_{n+1}|^2. The paper also derives applications to the 3D Hessian equation σ_2=1 and the 4D Hessian quotient equation σ_3=σ_1 under weaker assumptions than in previous work.

Significance. If the proof is completed, this would settle an open critical case posed by Collins–Jacob–Yau and Li, and would strengthen the connection between the LYZ equation and Bridgeland stability. The uniform Hessian estimate with constants independent of (θ(t)−(n−2)π/2)^{-1} is an important technical step, and the reduction of the new Liouville theorem via the extra variable z_{n+1} is elegant and potentially useful beyond this paper. The applications improve earlier results by Sun and Székelyhidi. However, the proof of the Liouville theorem contains a load-bearing gap in its scaling normalization, and the most novel case is partly delegated to prior work; these issues need to be addressed before the main theorem is fully supported.

major comments (3)

[§4.3, Theorem 4.1, Case 1] The scaling normalization in the proof of Theorem 4.1 is not justified as stated. From the hypothesis ∥v∥_{C^1(C^n)}≤C, the function \tilde v(z)=C_1^{-2}(v(C_1 z)−inf v) with C_1=(sup v−inf v)^{1/2} has oscillation 1, but its gradient satisfies |∇\tilde v|≤C C_1^{-1}. The displayed condition |∇\tilde v|≤C_1 would require (sup v−inf v)≥C^2, which is not a consequence of the hypotheses. Since the subsequent argument in Case 2 uses the normalized bounds 0≤v≤1 and |∇v|≤1 (e.g. in the 'following [10,38]' step and in the Cartan-type lemma), this gap is load-bearing for the Liouville theorem, and Theorem 4.1 is used in §4.4 to rule out the nonconstant blow-up limit when 0<a_0<∞. A possible repair is to choose the scaling parameter λ=max(C,(sup v−inf v)^{1/2}), which gives |∇v_λ|≤1 and 0≤sup v_λ−inf v_λ≤1, and then rework the argument with those bounds; but as written the proof is incomplete.
[§4.3, Case 1] The proof of Case 1 is only sketched by 'Following the arguments in [38] and [10], we obtain...'. Since Theorem 4.1 is a new Liouville theorem and Case 1 is an essential part of its proof, this delegation is not adequate. The authors should either supply the full construction of v_∞, the verification that it is independent of z_n, and the contradiction with (53), or state and prove a precise lemma that covers this case with all hypotheses checked.
[Theorem 1.1 / §4.5] Theorem 1.1 asserts uniqueness of the smooth solution with sup_M u=0, but the proof in §4.5 only establishes existence via a subsequence limit. No uniqueness argument or reference is given. This can likely be fixed by a standard maximum principle: at a maximum of u_1−u_2 one has χ_{u_1}≤χ_{u_2}, whence θ(χ_{u_1})≤θ(χ_{u_2}); equality of the phases then forces equality of the Hermitian matrices and hence u_1−u_2 constant. The authors should include this argument or an explicit reference.

minor comments (4)

[Corollary 1.1] The statement 'χ^2∧ω>0 as a positive (2,2)-form' seems to be a typo: in dimension 3 χ^2∧ω is a (3,3)-form, while the correct subsolution condition appears in Corollary 5.1 as χ∧ω>0 as a (2,2)-form. Please harmonize.
[Lemma 4.2] In the comparison principle, the displayed '≤−ϵn<0' after expansion is not literally correct: the expansion contains nonnegative lower-order terms whose coefficient is not simply ϵn. The contradiction still follows because w∈Γ_{n−1} makes the perturbed inequality strictly smaller than the subsolution inequality, but the formula should be corrected.
[§4.4, Eq. (60)] The exponent in the Hölder estimate '|\hat u_{t_i}|_{C^{1,2/3}}≤C' is unusual; please clarify whether this follows from Schauder estimates with the available L∞ bounds on Δ\hat u_{t_i} and the C^0 bound, and state the relevant standard result.
[§5.2] Typo: '4-dimesional' should be '4-dimensional'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the existence proof rests on external theorems (Lin, Sun, Collins–Jacob–Yau) plus new a priori estimates, not on the theorem being proved.

full rationale

The derivation chain is not circular. The paper reduces the critical LYZ equation to the supercritical approximating family (6), whose solvability is quoted from Lin (arXiv:2310.05339) and Sun, both external. The uniform C^0 bound is quoted from Collins–Jacob–Yau [7]. The new content is the Hou–Ma–Wu-type Hessian estimate and the gradient estimate by blow-up, whose new ingredient is the Liouville theorem for sigma_{n-1}+sigma_n=0. No fitted parameter is renamed as a prediction, and the target solution is not built from itself. The only self-citation, [12], supplies standard linearized-operator formulas (16)-(17) and is not load-bearing for the main existence argument. The scaling normalization in (52) and the sketched Case 1 of Theorem 4.1 are proof-completeness or correctness concerns, not circularity, since they do not reduce the theorem to its own assumptions. The manuscript is self-contained against external benchmarks for its central claim, so the appropriate circularity score is 0.

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

The paper introduces no free parameters or invented physical entities. The main external inputs are the supercritical existence theorem (Lin/Sun), standard potential-theoretic tools, and the known geometric setup of the LYZ equation. The proof of the critical case is a derivation from these ingredients.

assumptions (6)

domain assumption Existence of smooth solutions for the supercritical LYZ equations (Lin [27], see also Sun [37])
The proof of Theorem 1.1 uses Lin's theorem to obtain solutions u_t of the approximating supercritical equations (6). This is a black-box from a preprint/published alternative; if the supercritical theorem were false, the construction of u_t would fail.
domain assumption Subsolution condition (5) is an assumption of Theorem 1.1
This is a hypothesis of the main theorem, not derived within the paper.
standard math Im (χ_u+√−1ω)^{n−1} > 0 follows from the subsolution condition
Used in Lemma 2.3 to show the imaginary part of the total integral is positive; cited from Collins–Jacob–Yau [7].
standard math Schur–Horn theorem (Lemma 3.3)
Used to compare Σ F^{īi}(χ_u)_{īi} with Σ F^{īi}μ_i, where μ_i are eigenvalues of χ_u.
standard math Liouville theorems of Dinew–Kołodziej for σ_n=0 and σ_{n−1}=0 on C^n
Used in Case 1 and Subcase 2.1 of the gradient estimate; cited from [10].
standard math Standard elliptic regularity and C^{2,α} estimates from Collins–Jacob–Yau [7]
Used at the end of the proof of Theorem 1.1 to pass from uniform estimates to a smooth limit.

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Cite this review

Pith. "Pith review of The Critical LYZ Equation in K\"ahler Geometry." pith.science (2026). https://pith.science/paper/BAU6TUXD

@misc{pith2026251121492,
  author       = {Pith},
  title        = {Pith review of: The Critical LYZ Equation in K\"ahler Geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BAU6TUXD}},
  note         = {Machine review of arXiv:2511.21492}
}

abstract

We establish the existence of smooth solutions for the LYZ equation at the critical phase $\theta =(n-2)\frac{\pi}{2}$, thereby solving the critical case of a problem posed by Collins-Jacob-Yau and Li concerning the solvability for phase $\theta \leq (n-2)\frac{\pi}{2}$. As applications, we solve the 3D Hessian equation $\sigma_2 = 1$ and the 4D Hessian quotient equation $\sigma_3 = \sigma_1$ under weaker assumptions than previously required.

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