{"id":"08a4e862-a3a3-46ea-a546-9b22b81969ea","arxiv_id":"2608.01198","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves the Datar-Mete-Song conjecture: a pair of Kähler classes is semi-stable exactly when its minimal J-slope equals the topological J-slope.","lead":"A mathematics paper proves a conjecture about when a geometric equation, the J-equation, has solutions on Kähler manifolds: stability of a pair of shapes is exactly detected by a minimal slope number. The proof gives geometers a numerical way to detect where solutions break down and, on toric spaces, where weak solutions remain smooth.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The semi-stable direction of Theorem 1 depends on an unstated envelope-regularity theorem from [23] via Lemma 4; if that theorem's hypotheses or formula (1.4) do not cover this setting, the proof of ζ_min(α,β)=μ fails as written.","rationale":"The central claim is the full Datar–Mete–Song characterization of semi-stability by minimal slope. Part (1) is the hard direction; part (2) is a blow-up computation that checks out. The only step where the argument leaves the framework of the paper is Lemma 4's use of [23]. Everything else—the envelope construction in Lemma 3, the convergence in Lemma 2, and the pointwise linear-algebra inequality—is either proved in the text or easily patched. In particular, the diagonalization in Lemma 4 is not a real flaw: even with off-diagonal terms in Θ, the wedge product with diagonal T_ac only sees the diagonal entries of Θ, and the trace hypothesis gives their sum equal to n. The potential fragility is entirely the imported regularity and support formula for ⟨T^n⟩. This matches the reader's weakest assumption. A conditional verdict is appropriate: the proof is valid if [23] applies as stated, but it is not fully verified as written without checking that theorem. No change to the reader's verdict is needed.","tokens_in":12774,"tokens_out":15343,"duration_ms":176624,"concrete_test":"Obtain [23] and verify Theorem 1.1 and equation (1.4) verbatim. Specifically check: (a) whether the theorem permits θ to be a smooth semipositive form with [θ] big (rather than Kähler), γ to be a smooth representative of a pseudoeffective class D, and [θ]−[γ] to be big and nef; (b) whether the formula gives MA(P_θ[φ]) = 1_{P=φ} θ^n on Amp([θ])\\Sing(φ). If both hold, Lemma 4's mechanism is sound; if not, re-run the proof of Theorem 1(1) with the actual formula to see whether the inequality ∫Θ∧⟨T^{n-1}⟩≥∫⟨T^n⟩ still follows.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing point in Theorem 1(1) is Lemma 4, which imports [23, Thm 1.1 and (1.4)] to conclude two things about u=P_θ[φ_D](0): (i) u has locally bounded real Hessian on Amp(L)\\SuppD, and (ii) ⟨T^n⟩|_U = 1_{u=0} θ^n|_U for U=Y\\E. These are not derived from the definitions in the paper; they are the entire mechanism that turns the pointwise inequality Θ∧T^{n-1}≥T^n on the contact set into the global inequality ∫Θ∧⟨T^{n-1}⟩≥∫⟨T^n⟩. The manuscript never states the exact hypotheses of [23], so one cannot check from the text that they are met when θ is only semipositive (pulled back from a Kähler class), [θ] is big, and [θ]−[θ_D] is big and nef. If [23] requires θ to be Kähler on all of Y, or if formula (1.4) has a different density, Lemma 4 and hence Theorem 1(1) fail as written. This is an imported-result risk, not an internal contradiction; the unstable direction in Theorem 1(2) is self-contained and appears sound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":13031,"tokens_out":14435,"duration_ms":157391,"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":[{"comment":"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.","section":"Section 2.1, Lemma 4"},{"comment":"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.","section":"Section 3, proof of Theorem 3(2)"}],"minor_comments":[{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"Lemma 3"},{"comment":"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.","section":"Proof of Theorem 1(1)"},{"comment":"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.","section":"Proof of Theorem 3(2)"},{"comment":"There is a typo in 'destablizing' which should read 'destabilizing'; the same typo appears in the statement of Lemma 7.","section":"Definition 2"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the unverified import in Lemma 4. I could not check from the submitted text whether [23] applies under the stated hypotheses; if it does not, Theorem 1(1) fails. The unstable direction and the toric analyticity proof are coherent and checkable. The proof of Theorem 3(2) also needs a global argument identifying the limit as a weak solution. Both issues appear repairable, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe short version: this paper resolves the Datar–Mete–Song minimal slope conjecture in the expected way, and the resolution looks mostly right. The semi-stable direction, however, rests on an imported envelope-regularity theorem that the paper never states precisely. That is the thing to check before you trust Theorem 1(1).\n\nWhat is actually new: Theorem 1 extends the Datar–Mete–Song characterization from projective polarized pairs to arbitrary Kähler classes, and the unstable direction (item 2) is proved with a clean blow-up argument. Lemma 5's intersection computations on the exceptional divisor are correct and self-contained. Lemma 2's reduction from R-divisors to Q-divisors is a useful tool. The toric analyticity result for the optimally destabilizing set is also new, and the proof is straightforward once you notice the torus action preserves the numerical classes.\n\nThe soft spots are real but not obviously fatal. Lemma 4 imports [23, Thm 1.1 and (1.4)] to conclude that the envelope u has locally bounded Hessian on Amp(L)\\SuppD and that the Monge–Ampère measure has density 1_{u=0}θ^n. That is the entire mechanism turning the pointwise inequality into the global slope inequality, but the hypotheses of [23] are not stated. The reader can't check whether θ semipositive (not Kähler) with big class and L big and nef is covered. If the theorem does not apply, Theorem 1(1) fails as written. This is an imported-result gap, not an internal contradiction. Also, the diagonalization in Lemma 4 is sloppy (you can't simultaneously diagonalize θ, Θ, and T), though the inequality is a standard one and can be fixed by diagonalizing only T with respect to θ. In Theorem 3(2), the limit of the smooth toric solutions is shown to be smooth on the big torus, but the proof never verifies that this limit satisfies the J-equation globally or identifies it with Murakami's weak solution. That is a missing step, not just a detail.\n\nWho this is for: people working on J-equation stability, degenerate complex Hessian equations, and toric methods. It deserves a serious referee. I would send it out and ask the author to state and verify the envelope theorem, fix the diagonalization, and complete the toric limiting argument. If those land, the paper is a solid contribution.\n\nBest,\n[You]","headline":"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.","tokens_in":13568,"tokens_out":3782,"would_cite":true,"duration_ms":42202,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32Q15","53C55","14M25","32U05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Kähler pair is semi-stable exactly when its minimal birational slope equals the topological J-slope μ, and is strictly smaller otherwise.","keywords":["J-slope","minimal slope","semi-stability","J-equation","envelope with prescribed singularities","non-pluripolar products","toric Kähler manifold","destabilizing subvarieties"],"falsifier":"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.","tokens_in":12543,"feed_emoji":"📐","tokens_out":12700,"duration_ms":119895,"temperature":0.7,"pith_summary":"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.","feed_headline":"A single slope infimum settles Kähler pair stability","feed_subtitle":"A pair of Kähler classes is semi-stable exactly when its minimal birational slope equals the topological J-slope.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $(\\mathbb{C}^*)^n$."],"supporting_citations":[{"why":"Defines the semi-stable condition and the minimal slope, states the conjecture (with proof in the surface case), and provides the baseline reference the paper completes.","marker":"[10]"},{"why":"Supplies the envelope-regularity theorem (locally bounded Hessian and the support formula for the Monge-Ampère measure) used in Lemma 4 to prove the key inequality.","marker":"[23]"},{"why":"Establishes the Nakai-Moishezon criterion for the $J$-equation on compact Kähler manifolds, used to produce Kähler forms $\\omega_\\varepsilon,\\chi_\\varepsilon$ for the perturbed stable pairs.","marker":"[29]"},{"why":"Provides the non-pluripolar product theory in big cohomology classes, used to define $\\langle T^n\\rangle$ and $\\langle T^{n-1}\\rangle$ and equate them with cup products when the class is nef.","marker":"[1]"},{"why":"Gives the resolution result for Kähler classes on modifications, used to replace $\\mathbb{R}$-divisors by $\\mathbb{Q}$-divisors and to construct big and nef test classes.","marker":"[34]"},{"why":"Supplies the numerical characterization of the Kähler cone used to conclude that the test class $L_t$ is big in the unstable direction.","marker":"[13]"},{"why":"Provides the support theorems for closed currents on analytic sets, used to compute intersection numbers with the exceptional divisor in the destabilizing blow-up.","marker":"[12]"},{"why":"Proves convergence of the $J$-flow on toric manifolds, invoked to obtain smooth convex potentials and Hessian bounds on the big torus.","marker":"[8]"},{"why":"Establishes existence of a weak solution of the $J$-equation in the semi-stable case, which the paper then shows is smooth on the big torus in the toric invariant case.","marker":"[25]"}],"fun_headline_variants":["Minimal J-slope criterion proves Datar-Mete-Song","Semi-stability iff minimal slope equals topological J-slope","Infimum of birational slopes decides Kähler stability","J-slope semi-stability pinned by minimal birational slope"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Minimal J-slope criterion proves Datar-Mete-Song","Semi-stability iff minimal slope equals topological J-slope","Infimum of birational slopes decides Kähler stability","J-slope semi-stability pinned by minimal birational slope"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001199,"raw_usage":{"total_tokens":4951,"prompt_tokens":960,"completion_tokens":3991,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":576,"completion_tokens_details":{"reasoning_tokens":3920}},"tokens_in":576,"tokens_out":3991,"duration_ms":24802,"temperature":1.0,"reasoning_tokens":3920,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T04:14:17.815572+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"and Datar, V.,Minimal slopes and bubbling for complex Hessian equations, Adv","cited_arxiv_id":null,"evidence_quote":"Defines the semi-stable condition and the minimal slope, states the conjecture (with proof in the surface case), and provides the baseline reference the paper completes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the envelope-regularity theorem (locally bounded Hessian and the support formula for the Monge-Ampère measure) used in Lemma 4 to prove the key inequality."},{"cited_title":"and Zeriahi, A.,Monge–Amp` ere equations in big cohomology classes, Acta Math","cited_arxiv_id":null,"evidence_quote":"Provides the non-pluripolar product theory in big cohomology classes, used to define $\\langle T^n\\rangle$ and $\\langle T^{n-1}\\rangle$ and equate them with cup products when the class is nef."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the resolution result for Kähler classes on modifications, used to replace $\\mathbb{R}$-divisors by $\\mathbb{Q}$-divisors and to construct big and nef test classes."},{"cited_title":"and P˘ aun, M.,Numerical characterization of the K¨ ahler cone of a compact K¨ ahler manifold, Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the numerical characterization of the Kähler cone used to conclude that the test class $L_t$ is big in the unstable direction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the support theorems for closed currents on analytic sets, used to compute intersection numbers with the exceptional divisor in the destabilizing blow-up."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves convergence of the $J$-flow on toric manifolds, invoked to obtain smooth convex potentials and Hessian bounds on the big torus."},{"cited_title":"Weak solutions of the generalized Monge-Amp\\`ere equation and the supercritical deformed Hermitian-Yang-Mills equation: boundary cases","cited_arxiv_id":"2605.29258","evidence_quote":"Establishes existence of a weak solution of the $J$-equation in the semi-stable case, which the paper then shows is smooth on the big torus in the toric invariant case."}],"review_version":2}