{"id":"cff3f6e4-e002-4c7a-ae6f-ecba86b187aa","arxiv_id":"2501.04910","paper_version":6,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Slope stability of reflexive sheaves with respect to big classes is equivalent to existence of T-Hermitian-Einstein metrics when the big class has a birational Zariski decomposition with semiample positive part.","lead":"This paper extends a famous bridge between algebraic stability and curvature conditions from Kaehler manifolds to 'big' cohomology classes on singular complex varieties. The main payoff is that stability and special metrics remain equivalent for canonical models of general type, a setting central to classification geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6.1's HE⇒stability direction assumes the T-HE metric lives on the Zariski-resolution µ, while Definition 5.4 only gives it on some resolution; the transfer needs the very resolution-independence (Lemma 5.5/Assumption 3.1) the theorem claims to avoid.","rationale":"I read the main claim as the full equivalence stated in the abstract and Theorem 1.5, even though Theorem 6.1's statement prints only the \"if\" direction. The proof tries both directions; the first (\"stable ⇒ HE\") also uses Lemma 4.7, which again uses Assumption 3.1, but at least for stability the definition quantifies over all resolutions, so the chosen µ is covered once the definition is accepted. The sharper problem is the converse: Definition 5.4 only gives existence on some resolution. The proof silently upgrades to the Zariski-resolution µ. This is not a stylistic shortcut; it is the content of Lemma 5.5. Since Lemma 5.5 is proved under Assumption 3.1 and Theorem 6.1 claims to drop it, the missing piece is an explicit verification that the exceptional-divisor orthogonality holds in the semiample-positive-part case. The natural verification is plausible and probably true: if µ^*α = P + D and P is nef/semiample big, then for any further resolution the positive part is the pullback of P and the negative part is the pullback of D, so the problematic intersections vanish by the projection formula. But the paper does not include this argument at the point where it is needed. The reader's weakest_assumption identified the same region; I would sharpen it to this specific unproved transfer in the second paragraph of Theorem 6.1's proof. Because the missing step is fillable rather than demonstrably false, CONDITIONAL is the right level; I do not move the verdict.","tokens_in":22046,"tokens_out":12138,"duration_ms":117233,"concrete_test":"Take the setting of Theorem 6.1. Let µ: Z → X be the resolution realizing µ^*α = P + D with P semiample big and D effective µ-exceptional, and let η: V → X be another resolution. Choose a common resolution W with p: W → V, q: W → Z, and compute the negative part of p^*η^*α = q^*µ^*α using Lemma 2.21. Verify the orthogonality identity ⟨(p^*η^*α)^(n−1)⟩·[E] = 0 for every p- and q-exceptional divisor E, and then carry out the pushforward of a T-HE metric from V to Z as in Lemma 5.5. If the identity fails, or if it requires P to be semiample rather than merely nef, the proof of Theorem 6.1 is incomplete and the verdict stays CONDITIONAL.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the Kobayashi–Hitchin equivalence in Theorem 6.1. The direction that is actually load-bearing is \"E admits a T-HE metric ⇒ E is α^(n−1)-stable\", and it is not proved as stated. Definition 5.4 defines \"E admits a T-HE metric\" existentially: there is some resolution η: V → X and a current T_V ∈ η^*α such that η^*E is T_V-HE. The proof of Theorem 6.1, however, begins the converse direction with \"assume that µ^*E admits a T := (π^*ω + [D])-HE metric\", where µ: Z → X is the particular resolution realizing the birational Zariski decomposition. Replacing η by µ is exactly the resolution-independence statement of Lemma 5.5, whose proof explicitly invokes Assumption 3.1 (exceptional-divisor orthogonality). The theorem says Assumption 3.1 is not needed, but the paper does not prove the needed orthogonality for the resolutions appearing in Lemma 5.5. The natural repair — write µ^*α = P + D with P big semiample, and observe that for any further pullback the negative part is the pullback of D, so exceptional intersections vanish by the projection formula — is not carried out. Without that check, the normal-space definitions of stability and HE-metrics may depend on the chosen resolution, and the proof of the correspondence has a genuine gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the notions of slope stability and Hermitian-Einstein (HE) metrics from Kähler classes to arbitrary big cohomology classes on compact normal complex spaces, using the non-pluripolar product ⟨α^{n-1}⟩. The central result, Theorem 1.5/6.1, is a Kobayashi–Hitchin correspondence for reflexive sheaves on compact normal spaces when the big class admits a birational Zariski decomposition with semiample positive part. The proof strategy is to establish bimeromorphic invariance of stability and of the existence of HE metrics under an orthogonality assumption (Assumption 3.1), then descend to a Kähler class on a model via [Chen25]. A further result, Theorem 1.6/Proposition 7.7, proves the Bogomolov–Gieseker inequality for nef and big classes, with an equality characterization on minimal projective varieties of general type.","tokens_in":22445,"tokens_out":8018,"duration_ms":73228,"significance":"If correct, the main theorem would be a meaningful extension of the Kobayashi–Hitchin correspondence to singular spaces and big classes, with applications to canonical models, tangent sheaf polystability, and the Bogomolov–Gieseker inequality. The paper has a clear overall strategy and makes use of substantial external results (Chen25, BEGZ10, Nyström, BCHM10). The Bogomolov–Gieseker section is mostly self-contained and contains a useful openness argument. However, the proof of the main correspondence has a load-bearing gap concerning resolution independence under the stated hypotheses, and the formal statement of Theorem 6.1 is weaker than the abstract and Theorem 1.5. These issues are local and fixable, but they currently prevent the main claim from being established as written.","major_comments":[{"comment":"The statement of Theorem 6.1 proves only one direction: it says 'E is ⟨α^{n-1}⟩-stable if E admits a T-HE metric'. The abstract and Theorem 1.5 assert the full equivalence. Moreover, the proof of the converse direction begins with 'assume that µ^*E admits a T := (π^*ω + [D])-HE metric' on the specific resolution µ of the birational Zariski decomposition, whereas Definition 5.4 defines 'E admits a T-HE metric' existentially over some resolution. Passing from the given resolution to µ is exactly the resolution-independence statement of Lemma 5.5, whose proof invokes Assumption 3.1. Since Theorem 6.1 claims that Assumption 3.1 is not needed, this step is not justified. The paper must either prove the required orthogonality for the resolutions involved or replace Definition 5.4 with a resolution-choice that is compatible with the theorem.","section":"§6, Theorem 6.1 and Definition 5.4"},{"comment":"Lemma 4.7, which establishes independence of the chosen resolution in Definition 4.6, is proved under Assumption 3.1. The proof of Theorem 6.1 repeatedly applies Lemma 4.7 and Theorem 4.9 to conclude stability of µ^*E from stability of E, and vice versa, without Assumption 3.1. In the special case of a birational Zariski decomposition with semiample positive part, the needed orthogonality ⟨(µ^*α)^{n-1}⟩·[D] = 0 for π-exceptional divisors D may follow from the projection formula because ⟨µ^*α⟩ = π^*ω and D is π-exceptional, but this argument is not written. As it stands, the theorem's claim that Assumption 3.1 is unnecessary is not supported by the proof.","section":"§4.2, Lemma 4.7 and §6 proof"},{"comment":"The transition from the ω-HE metric on π[∗]µ[∗]E, obtained from Chen25, to the T-HE metric on µ^*E with T = π^*ω + [D] uses Theorem 5.6, a statement whose proof relies on Assumption 3.1 (via equation (4.3)). The special case D being π-exceptional and π^*ω being semiample should make the check feasible by the projection formula, but the verification of the Einstein constant and the integrability condition ∫ |F|_T^2 T^n < ∞ is not carried out in the proof of Theorem 6.1. This is another load-bearing point that needs an explicit argument.","section":"§5.3 and §6 proof"},{"comment":"Definition 4.6 defines ⟨α^{n-1}⟩-stability on a normal space by requiring stability of the reflexive pullback for every resolution. Lemma 4.7 then proves that this is equivalent to checking one resolution, under Assumption 3.1. Theorem 6.1 states that for classes with a birational Zariski decomposition with semiample positive part, Assumption 3.1 is unnecessary, but the paper does not prove that Definition 4.6 is well-posed for such classes without the assumption. The reader is left unable to determine whether the main theorem is about Definition 4.6 at all, or about a weaker notion where only the chosen resolution is tested.","section":"§4.2, Definition 4.6"}],"minor_comments":[{"comment":"There are several typographical inconsistencies: 'Kobayashi-Hichin' should be 'Kobayashi-Hitchin', and 'Kähler' is misspelled in a few places (e.g., 'compact Kähler manifolds' appears as 'compact K¨ aher manifolds' in §1.2).","section":"§1.2, Abstract"},{"comment":"Definition 5.4 reads 'We fix a resolution π:Y→X' but the definition is existential in the resolution. The wording should be changed to avoid the apparent contradiction.","section":"§5.2, Definition 5.4"},{"comment":"The proof writes 'p2_*(p1^*h)' as the transferred metric; a pushforward of a Hermitian metric is not a standard operation and needs a precise definition, especially regarding regularity on the ample locus.","section":"§5.2, Lemma 5.5 proof"},{"comment":"The displayed formula in (7.5) contains several typos: 'T r(bp · F_{ν^*h0} · bp + \\bar∂ bp ∧ ∂_{h0} bp)' should presumably be 'tr(bp · F_{ν^*h0} · bp + \\bar∂ bp ∧ ∂_{h0} bp)', and the following inequality uses '∥Fh0∥L∞rk(F)' which should be 'rk(F)∥F_{h0}∥_{L∞}'.","section":"§7.1, Lemma 7.2 proof"},{"comment":"The notation π[∗]E is used in Example 4.5 before it is formally defined in Lemma 4.7; a short definition in §4.1 would improve readability.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is quite rough and the main theorem's statement is internally inconsistent (Theorem 1.5 vs Theorem 6.1). The core idea is worth pursuing, but the proof as written has a genuine gap in the resolution-independence step. I would encourage the editor to send the paper back for a careful revision rather than reject, since the gap appears fixable by adding the missing orthogonality argument or by restricting the statement to the specific resolution used in the Zariski decomposition."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version. The genuinely new things are the definitions of slope stability and T-Hermitian-Einstein metrics for big classes on normal spaces, plus the bimeromorphic invariance theorems under Assumption 3.1. The main Kobayashi-Hitchin claim is a reduction to Chen's theorem rather than a new analytic argument, and that reduction has a real gap in one direction.\n\nThe good: the positive-product formalism is used correctly; the invariance statements are properly qualified by Assumption 3.1; the discussion of that assumption, including where it is known (projective manifolds, Zariski-decomposable classes), is useful; and the Bogomolov-Gieseker inequality for nef and big classes is a sensible payoff. If you only need the bimeromorphic invariance under Assumption 3.1, this part is solid.\n\nThe problem is Section 6. Theorem 6.1 is supposed to drop Assumption 3.1 and prove the correspondence for big classes admitting a birational Zariski decomposition with semiample positive part. Theorem 1.5 states the full equivalence, but Theorem 6.1's statement only gives 'T-HE metric ⇒ stability,' and the proof of that direction treats only the specific resolution µ that realizes the Zariski decomposition, with the specific current π*ω + [D]. Definition 5.4 makes the hypothesis existential: some resolution, some current. Bridging that gap is exactly what Lemma 5.5 does, and Lemma 5.5's proof invokes Assumption 3.1. The theorem claims the assumption is unnecessary, but the needed orthogonality is not proved in the Zariski-decomposition setting. I think the fix is straightforward—pull everything back to a common resolution and use the projection formula—but it is absent. As written, the main theorem is not established. The statement mismatch between Theorems 1.5 and 6.1 is minor by comparison; the proof clearly intends both directions.\n\nWho this is for: people working on stability with respect to movable or big classes, or on singular varieties with big canonical classes. The definitions and invariance theorems are likely to be cited. The headline correspondence should be cited with care until the gap is closed.\n\nRecommendation: send it to a serious referee. The framework is useful and the gap looks repairable. The referee should ask for a complete proof of the HE-to-stability direction, specifically the resolution-independence without Assumption 3.1, and a consistent statement of the main theorem.","headline":"A credible framework paper for big-class stability and HE metrics, but the main Kobayashi-Hitchin correspondence is not proved as stated because the HE-to-stability direction needs resolution independence the paper does not supply.","tokens_in":22887,"tokens_out":6127,"would_cite":true,"duration_ms":54962,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J60","32Q26","14E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Slope stability and Hermitian-Einstein metrics are equivalent for big classes on normal spaces.","keywords":["Kobayashi-Hitchin correspondence","big cohomology class","slope stability","Hermitian-Einstein metric","non-pluripolar product","normal variety","birational Zariski decomposition","Bogomolov-Gieseker inequality"],"falsifier":"Produce a compact normal space X with a big class $\\alpha$ admitting a birational Zariski decomposition with semiample positive part and a resolution pi: Y to X with an exceptional divisor D such that angle-bracket (pi^*$\\alpha$)^{n-1} cdot [D] is nonzero; this would invalidate the resolution-independence step and with it Theorem 6.1. Equivalently, find a reflexive sheaf E on such an X that is slope stable with respect to one resolution but not another, or that is slope stable but provably admits no T-Hermitian-Einstein metric.","tokens_in":21853,"feed_emoji":"📏","tokens_out":7509,"duration_ms":61395,"temperature":0.7,"pith_summary":"This paper extends the Kobayashi-Hitchin correspondence—the equivalence between slope stability and the existence of Hermitian-Einstein metrics—from Kaehler classes to big cohomology classes on compact normal complex spaces. The main theorem states that when a big class admits a birational Zariski decomposition with semiample positive part, a reflexive sheaf is slope stable with respect to the class if and only if it carries a Hermitian-Einstein metric adapted to a suitable closed positive current in the class. The proof works by transferring the statement along a birational map to a model where the positive part becomes a genuine Kaehler class, where the classical correspondence is known, and then pulling the metric and stability back. A corollary gives the correspondence for projective varieties of general type, and the same techniques yield a Bogomolov-Gieseker inequality for slope-stable reflexive sheaves with respect to nef and big classes.","feed_headline":"Stability and Einstein metrics coincide for big classes","feed_subtitle":"The Kobayashi-Hitchin correspondence now covers big cohomology classes with semiample positive part on singular spaces.","key_machinery":"The load-bearing object is the positive product angle-bracket $alpha^{{n-1}}$ of a big class, defined via non-pluripolar products of closed positive currents with minimal singularities. It replaces the ordinary wedge power when the class is not nef, and it is used to define both the slope mu_alpha(E) and the Einstein constant $\\lambda$. The second key mechanism is bimeromorphic invariance: under an orthogonality assumption (Assumption 3.1), both slope stability and the existence of T-Hermitian-Einstein metrics are unchanged under $\\beta$-negative birational maps, which allows the problem to be pushed to a resolution or a canonical model where $\\alpha$ becomes a pullback of a Kaehler class. The birational Zariski decomposition with semiample positive part is what guarantees such a model exists without needing the orthogonality assumption in the main theorem. Finally, the openness of slope stability in the class (Propositions 7.3 and 7.5) supplies the limit argument that yields the Bogomolov-Gieseker inequality for nef and big classes.","core_discovery":"The central claim is Theorem 6.1: let X be a compact normal space, $\\alpha$ a big Bott-Chern class, and E a reflexive sheaf. If $\\alpha$ admits a birational Zariski decomposition whose positive part is big and semiample, then E is angle-bracket $alpha^{{n-1}}$-slope stable exactly when it admits a T-Hermitian-Einstein metric for some closed positive (1,1)-current T in $\\alpha$. The slope is computed against the positive product angle-bracket $alpha^{{n-1}}$, defined through non-pluripolar products of currents with minimal singularities, and the metric is required to satisfy the Einstein condition $\\sqrt$(-1) Lambda_T F_h = $\\lambda$ Id on the ample locus away from the singularities, with finite $L^{2}$ norm. The proof shows resolution-independence of both notions under the stated hypothesis and uses the birational model where the positive part of the class is a pullback of a Kaehler class; on that model the result reduces to the known correspondence for normal varieties, and bimeromorphic invariance transfers it back. The same circle of ideas establishes the Bogomolov-Gieseker inequality for nef and big classes and the projective-flatness characterization in the equality case for minimal varieties of general type.","pith_inferences":["If the orthogonality condition in Assumption 3.1 holds for all big classes—a conjecture the paper connects to the differentiability of volumes of big classes—then the Kobayashi-Hitchin correspondence would extend to every big class on compact normal spaces, not only those with a birational Zariski decomposition with semiample positive part.","The limit argument in Section 7 suggests a stability threshold: for a fixed reflexive sheaf, the locus of big classes for which it is slope stable is open in the cone of big classes, so moduli spaces of sheaves with respect to big classes could be studied through their behavior near the boundary with nef classes.","The same birational-transfer strategy could apply to other geometric PDEs on singular spaces, such as Hermitian-Yang-Mills equations with additional twisted or Higgs fields, whenever a bimeromorphic model with a Kaehler class is available.","The projective-flatness conclusion in the equality case may be testable numerically: computing the discriminant Delta(E) cdot alpha^{n-2} on explicit examples of minimal varieties would verify the bound and its sharpness."],"forward_implications":["For any normal projective variety of general type with log terminal singularities, a reflexive sheaf is c_1(K_X)^{n-1}-slope stable if and only if it admits a T-Hermitian-Einstein metric.","The tangent sheaf, cotangent sheaf, and their tensor and exterior products of such a variety are slope polystable and admit T-Hermitian-Einstein metrics.","A slope-stable reflexive sheaf with respect to a nef and big class satisfies the Bogomolov-Gieseker inequality; in the equality case on minimal varieties of general type it is projectively flat on the ample locus.","On K3 surfaces, any nef and big class is semiample, so the equality case of the Bogomolov-Gieseker inequality forces projective flatness on the ample locus.","The correspondence and the inequality remain valid for polystable sheaves by the same arguments."],"supporting_citations":[{"why":"Supplies the Kobayashi-Hitchin correspondence on compact normal Kaehler spaces, the base case the paper reduces to.","marker":"[Chen25]"},{"why":"Defines non-pluripolar products and the positive product angle-bracket alpha^p, the machinery used to define slopes for big classes.","marker":"[BEGZ10]"},{"why":"Identifies the non-Kaehler locus with the null locus and proves orthogonality for classes admitting a Zariski decomposition, used in the proof of Theorem 6.1.","marker":"[CT15]"},{"why":"Proves the orthogonality condition on projective manifolds, giving the model case for Assumption 3.1.","marker":"[Nystr19]"},{"why":"Provides the existence of canonical models, making the birational Zariski decomposition concrete for varieties of general type.","marker":"[BCHM10]"},{"why":"Extends the correspondence to reflexive sheaves on Kaehler manifolds, the framework adopted here on singular spaces.","marker":"[BS94]"},{"why":"Establishes the classical Kobayashi-Hitchin correspondence on compact Kaehler manifolds, the origin of the result being generalized.","marker":"[Don87]"},{"why":"Gives the converse direction of the classical correspondence, the other half that the paper's main theorem extends.","marker":"[UY86]"}],"fun_headline_variants":["Kobayashi-Hitchin correspondence for big classes on singular spaces","Slope stability equivalent to Hermitian-Einstein for big classes","Big cohomology classes get Einstein metric criterion","Kobayashi-Hitchin correspondence for semiample positive parts","Einstein metrics detect slope stability for big classes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The main theorem assumes that for a class with a birational Zariski decomposition whose positive part is semiample, the positive product angle-bracket (pi^*$\\alpha$)^{n-1} pairs trivially with every exceptional divisor of every resolution of the space; if that orthogonality fails, the definitions of slope stability and Hermitian-Einstein metrics could depend on the resolution chosen and the correspondence would not be well-posed.","fun_headline_variants_meta":{"raw":{"variants":["Kobayashi-Hitchin correspondence for big classes on singular spaces","Slope stability equivalent to Hermitian-Einstein for big classes","Big cohomology classes get Einstein metric criterion","Kobayashi-Hitchin correspondence for semiample positive parts","Einstein metrics detect slope stability for big classes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000565,"raw_usage":{"total_tokens":2649,"prompt_tokens":889,"completion_tokens":1760,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":505,"completion_tokens_details":{"reasoning_tokens":1687}},"tokens_in":505,"tokens_out":1760,"duration_ms":11224,"temperature":1.0,"reasoning_tokens":1687,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:23:16.206336+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Produce a compact normal space X with a big class $\\alpha$ admitting a birational Zariski decomposition with semiample positive part and a resolution pi: Y to X with an exceptional divisor D such that angle-bracket (pi^*$\\alpha$)^{n-1} cdot [D] is nonzero; this would invalidate the resolution-independence step and with it Theorem 6.1. Equivalently, find a reflexive sheaf E on such an X that is slope stable with respect to one resolution but not another, or that is slope stable but provably admits no T-Hermitian-Einstein metric.","supporting_citations":[],"review_version":1}