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On the Datar-Mete-Song minimal slope conjecture

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Pith's one-line read A Kähler pair is semi-stable exactly when its minimal birational slope equals the topological J-slope μ, and is strictly smaller otherwise.

desk verdict Resolves the Datar–Mete–Song minimal slope conjecture in full Kähler generality, but the semi-stable direction leans on an unstated envelope-regularity theorem; worth a serious referee. read the letter →

arxiv 2608.01198 v2 pith:D3JHUHQG submitted 2026-08-02 math.AG math.APmath.CV

Xin Fu This is my paper · ORCID
classification math.AGmath.APmath.CV MSC 32Q1553C5514M2532U05
keywords J-slopeminimalslopesemi-stabilityJ-equationenvelopewithprescribedsingularitiesnon-pluripolarproductstoricKählermanifolddestabilizingsubvarieties
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 proves a conjecture that characterizes when a pair of Kähler classes is semi-stable in terms of a single numerical invariant, the minimal slope. For a pair $(\alpha,\beta)$ of Kähler classes on a compact Kähler manifold, semi-stability means that every proper subvariety $Z$ satisfies $d\,\alpha^{d-1}\cdot\beta\cdot Z \le \mu\,\alpha^d\cdot Z$, where $\mu$ is the topological $J$-slope; the minimal slope $\zeta_{\min}(\alpha,\beta)$ is the infimum of slopes of all birational test classes $L=\pi^*\alpha-[D]$ that are big and nef. The paper shows that $(\alpha,\beta)$ is semi-stable if and only if $\zeta_{\min}(\alpha,\beta)=\mu$. Because semi-stability is the borderline condition for solvability of the $J$-equation, this makes the stability threshold a computable intersection number rather than an analytic property. On toric manifolds the paper also proves that the set of optimally destabilizing subvarieties is a finite union of torus-invariant subvarieties, and that the weak solution of the $J$-equation is smooth on the dense big torus.

What carries the argument

The load-bearing object is the relative envelope $u=r+\varphi_D$, where $\varphi_D$ is the divisorial log potential of an effective divisor $D$ and $r$ is the $\theta_L$-psh envelope with prescribed singularities, i.e. the upper envelope of potentials $s$ with $\varphi_D+s\le 0$. A regularity theorem for such envelopes (imported from the literature) gives that $u$ has locally bounded real Hessian on $\mathrm{Amp}(L)\setminus \mathrm{Supp}\,D$ and that the Monge-Ampère measure of $T=\theta+i\partial\bar\partial u$ is supported on the contact set $\{u=0\}$, with density $\theta^n$ there. Because $u\le 0$ and $u=0$ on the contact set, the eigenvalues of $T$ at twice-differentiable contact points lie in $[0,1]$, so for any smooth semipositive $\Theta$ with $\mathrm{tr}_\theta\Theta=n$ the pointwise inequality $\Theta\wedge T_{\mathrm{ac}}^{n-1}\ge T_{\mathrm{ac}}^n$ holds after diagonalization. Integrating over $Y$ converts this pointwise comparison into the intersection inequality $n\,L^{n-1}\cdot\pi^*\beta\ge L^n$ that drives Theorem 1.

What would settle it

Exhibit any semi-stable pair $(\alpha,\beta)$ together with a birational modification $\pi:Y\to X$ and an effective $\mathbb{R}$-divisor $D$ for which $L=\pi^*\alpha-[D]$ is big and nef but $n\,L^{n-1}\cdot\pi^*\beta < \mu\,L^n$; the paper predicts no such data exist. A concrete place to test this is a toric surface with a semi-stable pair and $D$ supported on torus-invariant curves, where all intersection numbers are explicit.

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Extended reading notes

Core claim

The central discovery is that the infimum $\zeta_{\min}(\alpha,\beta)$ of slopes over all birational modifications $\pi:Y\to X$ and effective $\mathbb{R}$-divisors $D$ with $L=\pi^*\alpha-[D]$ big and nef is either exactly $\mu$ when the pair is semi-stable, or strictly smaller when the pair is unstable. Concretely, Theorem 1 proves the two directions: for a semi-stable pair, every admissible test satisfies $n\,L^{n-1}\cdot\pi^*\beta \ge \mu\,L^n$, so $\zeta_{\min}(\alpha,\beta)=\mu$; and for an unstable pair, a destabilizing subvariety $Z$ can be blown up to produce a test class whose slope is strictly below $\mu$, so $\zeta_{\min}(\alpha,\beta)<\mu$. The proof of the semi-stable direction goes by perturbing to a stable pair, solving the $J$-equation to obtain Kähler forms, and constructing an envelope current with divisorial singularities along $D$; the inequality then follows from a pointwise comparison on the contact set of the envelope.

Load-bearing premise

The semi-stable direction of the proof rests on an existing envelope-regularity result that is asserted to apply to a big and nef class with a divisorial potential having analytic singularities; if that theorem does not hold under these exact hypotheses, the key inequality in Lemma 4, and with it the whole equivalence, falls apart.

Editorial extensions

If this is right

  • For a semi-stable pair, every birational test class has slope at least the topological $J$-slope, so the minimal slope $\zeta_{\min}$ is a genuine numerical invariant of the pair and equals $\mu$.
  • An unstable pair is detected by an explicit destabilizing blow-up: blowing up a destabilizing subvariety and subtracting a small multiple of the exceptional divisor gives a test class with slope strictly below $\mu$.
  • Semi-stability of a Kähler pair can now be checked by intersection numbers alone, bypassing the analysis of the $J$-equation, in both the projective and Kähler settings.
  • On toric Kähler manifolds, the set of optimally destabilizing subvarieties is a finite union of torus-orbit closures, hence an analytic subset, for arbitrary (not necessarily torus-invariant) Kähler classes.
  • In the torus-invariant case, the weak solution of the $J$-equation in the semi-stable case is smooth and Kähler on the big torus $(\mathbb{C}^*)^n$.

Reading between the lines

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

  • The same envelope-comparison strategy may apply to other fully nonlinear equations whose stability thresholds are governed by the positivity of intersection numbers, such as the deformed Hermitian-Yang-Mills equation, although the paper does not state such an extension.
  • The analyticity of the optimally destabilizing set for general Kähler manifolds likely needs a mechanism beyond torus symmetry; the paper only proves the toric case.
  • Because the proof reduces the semi-stable direction to a pointwise inequality on the contact set of an envelope, the regularity of the weak solution outside the destabilizing set may be approachable by refining the envelope's Hessian bounds, a direction the paper explicitly leaves open.
  • The equality $\zeta_{\min}=\mu$ could serve as a practical numerical test for semi-stability in explicit examples, since computing slopes of torus-invariant test classes is a finite combinatorial problem on toric varieties.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proves the Datar-Mete-Song conjecture characterizing J-slope semi-stability by the minimal J-slope: for a semi-stable pair of Kähler classes (α, β) every birational test class has slope at least the topological J-slope, and for an unstable pair there is a test class with strictly smaller slope. The proof of the semi-stable direction uses a stable perturbation argument and a current with divisorial singularities, whose key estimates are imported from an envelope-regularity theorem of McCleerey. The unstable direction is based on blow-up intersection computations and appears self-contained. The paper also proves in the toric case that the union of optimally destabilizing subvarieties is analytic, and in the toric invariant case that Murakami's weak solution is smooth on the big torus.

Significance. If the results are correct, the paper settles Conjecture 1 of Datar-Mete-Song in full generality, going beyond the previously known surface and projective cases. The toric analyticity result for the optimally destabilizing locus is a substantial step beyond the two-dimensional case, and the partial regularity on the big torus is a natural continuation of Murakami's existence theorem. The unstable direction and the toric cycle-compactness argument are concrete and checkable. The main uncertainties are the unverified imported regularity result in Lemma 4 and the missing identification of the limiting current in Theorem 3(2); both are load-bearing but appear repairable.

major comments (2)
  1. [Section 2.1, Lemma 4] The proof of Lemma 4 imports [23, Theorem 1.1 and formula (1.4)] without stating the hypotheses of that theorem or verifying them for the present data. The manuscript asserts that the theorem applies directly because φ_D has analytic singularities, [θ] is big, [θ_D] is pseudoeffective, and [θ]−[θ_D] is big and nef, but it does not state the exact conditions under which [23] yields locally bounded real Hessian of u on Amp(L)\SuppD and the measure identity ⟨T^n⟩|_U = 1_{u=0}θ^n|_U. In particular, the manuscript only assumes that θ is Kähler on Y\E for an analytic set E containing SuppD and the complement of Amp(L), while θ is merely semipositive on all of Y. If [23] requires θ to be Kähler everywhere, or if formula (1.4) has a different normalization, then the identity ⟨T^n⟩|_U = 1_{u=0}θ^n|_U, and hence the global inequality in Lemma 4, fails as written. This identity is the entire mechanism converting the pointwise inequality on the contact set into the inequality ∫Θ∧⟨T^{n-1}⟩≥∫⟨T^n⟩, so the proof of Theorem 1(1) is incomplete unless the precise theorem is stated and its hypotheses are checked.
  2. [Section 3, proof of Theorem 3(2)] The proof constructs a subsequence of smooth solutions ψ_ε of the perturbed stable J-equations and shows that the limit ψ is smooth on the big torus. However, the proof never verifies that the limiting current T=ω+i∂∂̄ψ satisfies the global weak J-equation n⟨T^{n-1}⟩∧ω=μ⟨T^n⟩, nor that T coincides with Murakami's weak solution from [25]. Convergence of the approximating equations to the limiting equation is not automatic because non-pluripolar products are not continuous under L^1 convergence of potentials, and the C^{1,1} bounds are only local in the torus. Therefore the statement that 'there is a weak solution T of the J-equation' with the claimed smoothness does not follow from the given argument. The author must either prove directly that the limit satisfies the global weak equation, or appeal to a uniqueness theorem for the semi-stable weak solution and show that the constructed limit is that solution.
minor comments (5)
  1. [Abstract] The arXiv abstract and the full-text abstract are inconsistent: the former says the J-null locus is an analytic subset, while the latter says the set of optimally destabilizing subvarieties is finite. Theorem 3(1) proves analyticity of the union; finiteness of the collection of subvarieties is not proved. The terminology should be made consistent and precise.
  2. [Lemma 3] The hypothesis 'θ^n>0' is ambiguous: if interpreted pointwise it forces θ to be Kähler, which is false for the pullback forms θ=π^*ω_ε used in the proof of Theorem 1(1); if interpreted cohomologically it should be written as [θ]^n>0. Please clarify.
  3. [Proof of Theorem 1(1)] The sentence 'By lemma 2, one may assume that the divisor D has rational coefficient' is not literally justified by Lemma 2, which produces a sequence of rational divisors on a further modification with convergent slopes. The argument should instead apply the inequality to the rational approximations and then pass to the limit using the continuity of intersection numbers.
  4. [Proof of Theorem 3(2)] The final sentence 'the higher-order regularity follows by a standard argument' is too terse for a journal paper; the author should indicate which estimate (e.g., Evans-Krylov or Schauder) is used, given that the C^{1,1} bound is only local.
  5. [Definition 2] There is a typo in 'destablizing' which should read 'destabilizing'; the same typo appears in the statement of Lemma 7.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: Theorem 1 is proved by perturbing to the stable case and invoking external envelope-regularity theorems, not by assuming the conjecture.

full rationale

The paper does not define its target in terms of its inputs, fit any parameter and rename it a prediction, or lean on self-citations. The central semi-stable direction (Theorem 1(1)) reduces the desired inequality nL^{n-1}·π^*β ≥ μL^n to the stable perturbed case via Theorem 4 [29], constructs a current T with divisorial singularities (Lemma 3) using non-pluripolar products, and then obtains the key integration inequality in Lemma 4 from the external envelope-regularity theorem of McCleerey [23] and formula (1.4) there. That cited result is not by the present author and is not the Datar-Mete-Song conjecture; it is imported as an independent analytic theorem. The unstable direction (Theorem 1(2)) is self-contained: it exhibits a test divisor from the destabilizing subvariety and computes slope asymptotics via Lemma 5. There is no fitted input, no renaming of a known conjecture, and no load-bearing self-citation: references [10], [29], [25], and [23] are external, and the author's own work is not used as an assumption. Whether [23]'s hypotheses exactly cover semipositive θ and big-nef L is a correctness/verification concern, not a circularity.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no numerical parameters fitted to data and no new geometric entities such as particles, forces, or dimensions. It constructs currents and uses standard cohomological intersection numbers. All nontrivial inputs are published theorems in Kähler geometry, with the main unstated external dependence being McClereey's envelope regularity result.

assumptions (8)
  • standard math Song's Nakai-Moishezon criterion for the J-equation (Theorem 4).
    Used in Step 1 of Theorem 1(1) to obtain smooth solutions for perturbed stable pairs (α,β_ε). The theorem is imported, not proved.
  • standard math BEGZ non-pluripolar product theory in big cohomology classes.
    Used in Lemma 3 to assert that products of currents with minimal singularities represent ordinary cup products for nef classes.
  • standard math McClereey's envelope regularity theorem and formula (1.4).
    Used in Lemma 4 to get the Hessian bound and the contact-set support of ⟨T^n⟩. The paper's proof of Theorem 1(1) depends on this imported result.
  • standard math Bishop-Lieberman compactness for analytic cycles.
    Used in Lemma 7 to take limits of translated cycles while controlling their cohomology classes.
  • standard math Demailly-Paun numerical characterization of the Kähler cone.
    Used in Lemma 6 to conclude that the nef class L_t is big.
  • standard math Varouchas Grauert-Kähler class stability under blow-up.
    Used in Lemma 6 to produce the nef class π^*α-tF from the Kähler class on the blow-up.
  • standard math Collins-Szekelyhidi toric J-flow existence.
    Used in the proof of Theorem 3(2) to produce approximating smooth solutions for perturbed stable toric pairs.
  • standard math Rockafellar convex analysis theorem.
    Used in Theorem 3(2) to obtain local uniform boundedness of convex potentials on the big torus.

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Pith. "Pith review of On the Datar-Mete-Song minimal slope conjecture." pith.science (2026). https://pith.science/paper/D3JHUHQG

@misc{pith2026260801198,
  author       = {Pith},
  title        = {Pith review of: On the Datar-Mete-Song minimal slope conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D3JHUHQG}},
  note         = {Machine review of arXiv:2608.01198}
}
abstract

We prove a conjecture of Datar-Mete-Song \cite{DMS} characterizing $J$-slope semi-stability by the minimal $J$-slope. More precisely, for a semi-stable pair of K\"ahler classes $(\alpha,\beta)$ on a compact K\"ahler manifold $X$, every big and nef birational test class has slope at least the topological $J$-slope, whereas an unstable pair admits a test class with strictly smaller slope. We also introduce the $J$-null locus of a semi-stable pair and prove that it is an analytic subset of $X$ if $X$ is a compact K\"ahler surface or a compact toric K\"ahler manifold. In the toric invariant case, we show that Murakami's \cite{Murakami} weak solution to the $J$-equation is smooth and K\"ahler on the dense big torus $(\mathbb{C}^*)^n$ of $X$.

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