{"id":"6b1366b0-329c-4b9c-a44e-c190f2ce100b","arxiv_id":"2607.18699","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For compact normal analytic varieties in Fujiki's class, Miyaoka's inequality holds when the canonical divisor is nef, and the orbifold second Chern class is semipositive when the anti-canonical divisor is nef, under klt or related singularity hypotheses.","lead":"This paper proves Miyaoka-type Chern-class inequalities and semipositivity of the orbifold second Chern class for compact singular complex varieties in Fujiki's class, the class bimeromorphic to compact Kähler manifolds. The results extend tools used in the minimal model program beyond the projective and Kähler settings and may feed into abundance-type classification arguments.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 4.13(2), used in the proof of Theorem 1.3, is false as stated: O(1)⊕O(-1) on P^1×P^1 satisfies its hypotheses but has \\hat c_2 = -1.","rationale":"The paper's architecture is coherent and the external-input concern raised by the reader (Prop 4.5 / Lemma 4.6) is real, but I find a more immediate internal obstacle. Corollary 4.13(2) is stated for arbitrary reflexive sheaves and is used in the proof of Theorem 1.3. As written it is false: a direct smooth counterexample satisfies every hypothesis but violates the conclusion. The failure is in the proof's assertion that c1(E)·Ω≡0 implies α_1·Ω-semistability via Remark 3.6; Remark 3.6 is only an equivalence about numerical triviality of a homology class and has no bearing on destabilizing subsheaves. This is not a mere technicality: when -K_X·Ω=0, Theorem 1.3 lands exactly in this degenerate case. A repair is available—Prop 4.5 supplies μ_min≥0 for the polarization α_1·Ω, and since the average slope of T_X is 0, all Harder–Narasimhan slopes vanish, giving semistability—but this argument is absent and Cor 4.13 needs restatement. I therefore keep the CONDITIONAL verdict, with the required revision being Cor 4.13 and the degenerate case of Theorem 1.3.","tokens_in":24947,"tokens_out":17088,"duration_ms":135364,"concrete_test":"Check Cor 4.13(2) against X=P^1×P^1, E=O(1)⊕O(-1), Ω=1. Compute det E=O (nef), c1(E)=0, c1(E)^2·Ω=0, μ_min=0, but c2(E)=-1, so bc2(E)·Ω=-1<0. This directly disproves the corollary as stated. If the authors intend to restrict to E=T_X or Ω^1_X, the test should instead exhibit a klt Fano/Calabi–Yau variety with -K_X·Ω=0 but T_X not α_1·Ω-semistable; a first step is to search for a destabilizing subsheaf among known examples of singular Fano threefolds with a nef class H satisfying -K_X·H=0.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Corollary 4.13(2), used directly in the proofs of Theorems 1.1–1.3, is false as stated. Example: X=P^1×P^1, E=O(1)⊕O(-1), Ω=1. Then det E=O_X is nef, c1(E)=0, so c1(E)·Ω≡0 and c1(E)^2·Ω=0; the slope condition μ_min_{c1(E)·Ω}(E)≥0 is vacuous because all slopes are 0. Yet bc2(E)=c2(E)=-1<0. The proof's step 'E is α_1·Ω-semistable by Remark 3.6' is invalid: Remark 3.6 is a Hodge-index statement about numerical triviality of a class, not a semistability criterion. A reflexive sheaf with nef determinant and c1(E)·Ω=0 need not be semistable (witness O(1)⊕O(-1)). In Theorem 1.3, the case -K_X·Ω=0 is exactly the degenerate case where this false implication is invoked, so the proof of the main semipositivity theorem is incomplete. The gap is repairable if one instead uses Prop 4.5 with polarization α_1·Ω to force all Harder–Narasimhan slopes to vanish (since the average slope is 0 and μ_min≥0), but that argument is not in the paper and requires the stated hypotheses of Cor 4.13 to be strengthened.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies orbifold second Chern class inequalities for compact normal analytic varieties in Fujiki's class. It claims Miyaoka's inequality (3 c_hat2(X) - c_hat1(X)^2) · alpha_1 · ... · alpha_{n-2} >= 0 when K_X is nef, under canonical singularities or under quotient/rational singularities with X non-uniruled (Theorem 1.1), and a perturbed version for klt Kähler varieties (Theorem 1.2). It further claims semipositivity c_hat2(X) · alpha_1 · ... · alpha_{n-2} >= 0 when -K_X is nef and X is klt (Theorem 1.3), with Miyaoka-Yau type corollaries. The strategy combines recent generic nefness results for tangent/cotangent sheaves, an orbifold Grothendieck-Riemann-Roch additivity formula, a Hodge-index argument, and a Bogomolov-Gieseker inequality for mixed polarizations.","tokens_in":25308,"tokens_out":7297,"duration_ms":57626,"significance":"If the results hold, this is a substantial extension of Miyaoka's inequality and semipositivity of the second Chern class from projective and Kähler settings to singular varieties in Fujiki's class, with potential applications to abundance. The paper has several genuine strengths: it sets up orbifold Chern classes through orbifold modifications, proves an additivity statement via orbifold Grothendieck-Riemann-Roch, gives a Hodge-index framework for degenerations, and reduces the main inequalities to generic nefness plus Langer-type estimates. The significance is, however, conditional: the central semipositivity theorem depends on a corollary that is false as stated, and several key inputs are only cited to very recent preprints rather than proved. The paper also explicitly acknowledges that Corollary 1.4 was already established in [ZZZ25], which limits the novelty of that part.","major_comments":[{"comment":"Corollary 4.13(2) is false as stated. Take X = P^1 × P^1, E = O(1) ⊕ O(-1), Ω = 1 (n = 2). Then det E = O_X is nef, c1(E) = 0, so c1(E)^2 · Ω = 0 and c1(E) · Ω ≡ 0; the hypothesis μ_min_{c1(E)·Ω}(E) ≥ 0 is vacuous. But bc2(E) = c2(E) = -1, contradicting the claimed conclusion. The proof's step 'E is α_1·Ω-semistable by Remark 3.6' is invalid: Remark 3.6 is a Hodge-index statement about numerical triviality of a class, not a semistability criterion. Semistability with respect to α_1·Ω requires controlling slopes of all subsheaves, and c1(E)·Ω ≡ 0 does not imply that. This invalid step is used in the c1(K_X)^2·Ω = 0 case of Theorem 1.1 and in the degenerate case of Theorem 1.3. The gap may be repairable by using Proposition 4.5 with polarization α_1·Ω to force all Harder-Narasimhan slopes to vanish, but that argument is not present and needs a strengthened hypothesis.","section":"Corollary 4.13(2), proof of Theorems 1.1 and 1.3"},{"comment":"Theorem 1.3 rests on Proposition 4.5, whose proof is only a reduction to Lemma 4.6, and Lemma 4.6 is imported almost verbatim from [CP25, Theorem 3.1] with the sentence 'The proof of [CP25, Theorem 3.1] applies here with only minor modifications.' The existence of the holomorphic 2-form σ vanishing on general fibers, which is assumption (3) of Lemma 4.6, is asserted for rationally connected fibers only by reference. The production of a saturated positive-slope subsheaf with rationally connected leaves on non-Kähler Fujiki-class spaces also relies on [Ou25a, Theorem 1.4] and [CP25, Corollary 1.5]. Since these are recent preprints, the manuscript should either state them as precise external theorems or provide enough proof that the reader can verify the transfer to the singular non-Kähler setting. As written, the proof of the main -K_X nef theorem is conditional on these ingredients.","section":"Proposition 4.5 and Lemma 4.6"},{"comment":"Theorem 1.1(2) is stated as a separate theorem with different hypotheses (non-uniruled, quotient singularities in codimension 2, rational singularities), but its proof is dismissed with 'case (2) is analogous' and no details are given. Theorem 1.2, which is used for Corollary 1.4, also abbreviates the final estimate: after establishing μ_min ≥ 0, the proof says 'The remainder ... follows the same line of argument ... and we omit the details.' Since these are load-bearing parts of the paper's claims, the omitted arguments should be supplied or the theorems should be formulated as conditional statements with explicit references.","section":"Theorems 1.1(2) and 1.2"}],"minor_comments":[{"comment":"The content of Remark 3.6 is a Hodge-index identity, not a semistability criterion. Its later use in Corollary 4.13(2) is misleading and should be clarified or removed.","section":"Remark 3.6"},{"comment":"The proof asserts the existence of a positive orbifold (1,1)-form ω such that ω^{n-1}/(n-1)! = ω_1 ∧ ... ∧ ω_{n-1}; this is not automatic and should be justified or cited.","section":"Lemma 4.8"},{"comment":"The authors honestly note that Proposition 3.10 is not known for non-pluripolar products, which limits the psef extension. This limitation should be stated earlier, near Proposition 3.10, so readers do not over-interpret its scope.","section":"Section 5.1"},{"comment":"The reference [ZZZ25] lists 'Chuangjing Zhang'; if this is a typo for the author's name it should be corrected. Also, since [IJZ25] is a split version of the original preprint, the paper should state explicitly which results are proved here versus quoted from [IJZ25].","section":"References"},{"comment":"In the proof of Theorem 1.3, γ is set to c1(-K_X) · Ω and then Corollary 4.13 is applied to T_X. The notation should be aligned: Corollary 4.13 uses c1(E)·Ω, and for E = T_X this is c1(-K_X)·Ω, not c1(K_X)·Ω. The current wording could confuse the sign.","section":"Theorem 1.3 proof"}],"recommendation":"major_revision","confidential_remarks":"The novelty boundary needs editorial scrutiny. The authors themselves state that Corollary 1.4 was already proved in [ZZZ25], and Theorem 1.3 overlaps with [MWWZ25, Theorem 5.3] in the compact Kähler case. The main new content is the Fujiki-class, singular-variety extension and the Miyaoka inequality for nef K_X. The false Corollary 4.13(2) is a serious technical gap, though it appears repairable. I recommend major revision rather than rejection, but the revision should either prove the strengthened generic nefness argument or clearly restrict the statements to the cases where the current argument is valid."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take: the paper has a genuinely new and worthwhile statement — Miyaoka's inequality and semipositivity of the orbifold second Chern class for compact normal analytic varieties in Fujiki's class, not just for Kähler or projective varieties. The orbifold machinery and the Langer-style inequality are useful, and the exposition is honest about overlaps with [MWWZ25] and [ZZZ25]. But the proof of the main semipositivity theorem rests on a false step.\n\nThe stress-test note holds up. Corollary 4.13(2), used directly in Theorem 1.3, is false as stated. Take X = P^1×P^1, E = O(1)⊕O(-1), and Ω the point class. Then det E is trivial, so c1(E) = 0, c1(E)^2·Ω = 0, and the slope condition μ_min ≥ 0 is vacuous. Yet c2(E) = -1 < 0. The proof's line that \"E is α_1·Ω-semistable by Remark 3.6\" confuses a Hodge-index statement about numerical triviality with a semistability criterion. It is simply not true that a reflexive sheaf with nef determinant and c1(E)·Ω = 0 is semistable with respect to any polarization. This is not cosmetic: in Theorem 1.3, the case -K_X·Ω = 0 is exactly where this false implication is invoked.\n\nThe paper deserves credit for what is new: the singular Fujiki-class setting, the orbifold Bogomolov–Gieseker inequality for mixed polarizations, and the generic nefness results for tangent and cotangent sheaves. The structure is coherent and the overall strategy is recognizable from the MMP literature. But several load-bearing inputs are cited only to unpublished preprints ([Ou25a], [CP25], [MTTW25], [KO25], [ZZZ25]), some with author overlap; Theorem 1.1(2) is dismissed as analogous, and parts of Theorem 1.2 are omitted. I did not find a separate red flag in the stated inequalities themselves, but the proof is not complete as written.\n\nBottom line: the question is important and the paper should be sent to a serious referee, but not accepted in this form. The referee should ask for a corrected Corollary 4.13 (possibly by strengthening the hypotheses, e.g., requiring c1(E)·Ω to be a genuine positive polarization, or by handling the degenerate case via Prop 4.5 with a perturbation argument). Until that is fixed, I would not cite Theorem 1.3.\n\nBest.","headline":"Plausible and important extension of Miyaoka-type inequalities to singular Fujiki-class varieties, but Theorem 1.3 relies on a false corollary; the paper needs major revision before it can be trusted.","tokens_in":25824,"tokens_out":7775,"would_cite":false,"duration_ms":68232,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32J25","32Q15","14C30","14E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that compact normal analytic varieties in Fujiki's class with klt singularities and nef anti-canonical divisor have nonnegative orbifold second Chern class, and that Miyaoka's inequality holds when the canonical divisor is","keywords":["Miyaoka inequality","orbifold Chern classes","Fujiki class","klt singularities","generic nefness","Bogomolov-Gieseker inequality","nef anti-canonical divisor","complex analytic varieties"],"falsifier":"A compact klt variety X in Fujiki's class with −K_X nef and nef big classes α1,...,αn−2 for which the intersection number ĉ2(X)·α1·...·αn−2 is negative would directly refute Theorem 1.3; searching among singular compactifications with quotient singularities is the concrete check.","tokens_in":24820,"feed_emoji":"📐","tokens_out":5948,"duration_ms":63022,"temperature":0.7,"pith_summary":"The paper establishes Chern-class inequalities for singular complex analytic varieties that need not be Kähler, as long as they belong to Fujiki's class (bimeromorphic to compact Kähler manifolds). It proves Miyaoka's inequality—3 times the orbifold second Chern class minus the square of the orbifold first Chern class is nonnegative—when the canonical divisor is nef, and it proves semipositivity of the orbifold second Chern class when the anti-canonical divisor is nef. These are the first such results for non-Kähler singular spaces, and they extend a body of results previously known only for projective or Kähler varieties. The motivation is the minimal model program, where such Chern-class inequalities are known to feed into abundance-type conclusions.","feed_headline":"Nef anticanonical divisor forces semipositive orbifold c2","feed_subtitle":"Proof extends Miyaoka's inequality and Chern semipositivity to singular spaces in Fujiki's class, with klt singularities.","key_machinery":"The proof rests on two pillars. First, generic nefness theorems for tangent and cotangent sheaves: when K_X is nef the cotangent sheaf has nonnegative minimal slope, and when −K_X is nef the tangent sheaf does, using recent foliation-theoretic criteria. Second, an orbifold Bogomolov–Gieseker inequality for mixed polarizations, proved via orbifold Hermite–Einstein metrics and an orbifold Grothendieck–Riemann–Roch formula. The orbifold Chern classes are defined through bimeromorphic orbifold modifications, so they make sense for varieties with quotient singularities in codimension two, and they coincide with ordinary Chern classes where the variety is smooth in codimension two.","core_discovery":"The central result is Theorem 1.3: if X is an n-dimensional compact normal analytic variety in Fujiki's class with klt singularities and nef anti-canonical divisor, then for any nef and big classes α1,...,αn−2 in Bott–Chern cohomology, the orbifold second Chern class satisfies ĉ2(X)·α1·...·αn−2 ≥ 0. For the canonical-divisor side, Theorem 1.1 gives Miyaoka's inequality (3ĉ2(X) − ĉ1(X)^2)·α1·...·αn−2 ≥ 0 when K_X is nef and X has canonical singularities (or quotient singularities in codimension 2 with rational singularities). The paper also proves a version for klt Kähler varieties with a small perturbing polarization, and derives Miyaoka–Yau type inequalities from these results.","pith_inferences":["Editorial: the semipositivity result for singular non-Kähler spaces suggests that the numerical constraints behind abundance may extend beyond the Kähler setting; a direct test would be to see whether the equality structure theorem holds for klt Kähler varieties with nef anti-canonical divisor.","Editorial: the proof of generic nefness in the anti-canonical case is the main load-bearing imported input; making that step self-contained for singular Fujiki-class spaces would likely also yield the log-pair and psef generalizations discussed in the paper.","Editorial: the paper leaves open the psef case and explicitly notes that a quotient-bundle example shows generic nefness can fail when −K_X is merely psef; resolving the non-pluripolar product issue flagged in Section 5.1 would be the natural next step toward a Miyaoka inequality for psef canonical divisors.","Editorial: one can test Theorem 1.3 concretely by computing the orbifold second Chern class on explicit singular compactifications with quotient singularities in Fujiki's class and nef anti-canonical bundle; nonnegativity in such examples would corroborate the theorem, while a negative value would refute it."],"forward_implications":["On non-Kähler compact analytic varieties in Fujiki's class with nef canonical divisor and canonical singularities, Miyaoka's inequality holds against all nef and big classes.","On klt Kähler varieties with nef canonical divisor, Miyaoka's inequality holds after perturbing the polarization by a small multiple of a fixed Kähler class.","On klt varieties in Fujiki's class with nef anti-canonical divisor, the orbifold second Chern class is semipositive against all nef and big classes.","Combining with numerical dimension yields Miyaoka–Yau type inequalities for klt Kähler varieties with nef canonical or nef anti-canonical divisor.","If equality holds in Miyaoka's inequality on a compact Kähler manifold, the canonical divisor is semiample and the manifold is a complex torus, a torus fibration over a curve, or a product of a torus and a ball-quotient surface."],"fun_headline_variants":["Nef anticanonical forces semipositive orbifold c2 in Fujiki class","Miyaoka inequality and c2 semipositivity for singular varieties in Fujiki class","Orbifold c2 ≥ 0 under nef anticanonical on klt Fujiki spaces","Semipositive orbifold second Chern class from nef anticanonical"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The semipositivity theorem depends on the imported claim that the tangent sheaf is generically nef whenever the anti-canonical divisor is nef; if that generic-nefness statement fails for singular non-Kähler varieties in Fujiki's class, Theorem 1.3 breaks down before the orbifold Chern-class estimates are used.","fun_headline_variants_meta":{"raw":{"variants":["Nef anticanonical forces semipositive orbifold c2 in Fujiki class","Miyaoka inequality and c2 semipositivity for singular varieties in Fujiki class","Orbifold c2 ≥ 0 under nef anticanonical on klt Fujiki spaces","Semipositive orbifold second Chern class from nef anticanonical"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000909,"raw_usage":{"total_tokens":3707,"prompt_tokens":672,"completion_tokens":3035,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":416,"completion_tokens_details":{"reasoning_tokens":2938}},"tokens_in":416,"tokens_out":3035,"duration_ms":19112,"temperature":1.0,"reasoning_tokens":2938,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T14:36:40.566289+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A compact klt variety X in Fujiki's class with −K_X nef and nef big classes α1,...,αn−2 for which the intersection number ĉ2(X)·α1·...·αn−2 is negative would directly refute Theorem 1.3; searching among singular compactifications with quotient singularities is the concrete check.","supporting_citations":[],"review_version":1}