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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

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2026-08-01 14:36 UTC pith:L24NPJGQ

load-bearing objection 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. the 3 major comments →

arxiv 2607.18699 v1 pith:L24NPJGQ submitted 2026-07-21 math.AG math.CVmath.DG

Semipositivity of the orbifold second Chern class in Fujiki's class

Masataka Iwai , Satoshi Jinnouchi , Shiyu Zhang This is my paper
classification math.AG math.CVmath.DG MSC 32J2532Q1514C3014E30
keywords Miyaoka inequalityorbifold Chern classesFujiki classklt singularitiesgeneric nefnessBogomolov-Gieseker inequalitynef anti-canonical divisorcomplex analytic varieties
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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.

What carries the argument

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.

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Corollary 4.13(2), proof of Theorems 1.1 and 1.3] 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.
  2. [Proposition 4.5 and Lemma 4.6] 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.
  3. [Theorems 1.1(2) and 1.2] 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.
minor comments (5)
  1. [Remark 3.6] 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.
  2. [Lemma 4.8] 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.
  3. [Section 5.1] 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.
  4. [References] 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].
  5. [Theorem 1.3 proof] 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.

Circularity Check

0 steps flagged

No significant circularity: The main inequalities are derived from external generic-nefness and Bogomolov-Gieseker inputs, not from their own conclusions; overlapping-author citations are technical and non-load-bearing.

full rationale

The derivation chain is not circular. Theorem 1.3 applies Proposition 4.5 and Corollary 4.13 to the tangent sheaf; Proposition 4.5 is a generic-nefness statement proved by contradiction using Lemma 4.6, imported from the external Cao-Paun theorem [CP25, Thm 3.1], and it never assumes bc2(X)>=0. Corollary 4.13 follows from the Langer-type slope inequalities (Prop 4.12), the Hodge index theorem (Lemma 3.5 / Prop 3.7), and the orbifold Bogomolov-Gieseker inequality (Cor 4.9), whose inputs are Hermite-Einstein existence results from [ZZZ25, Thm 3.3] and [CW24], not the theorem being proved. Theorem 1.1 similarly combines the generic nefness of the cotangent (Prop 4.4) with Chern-class algebra. Corollary 1.4 is explicitly acknowledged as already contained in [ZZZ25, Thm 1.1] and is not used to prove Theorems 1.1-1.3. The self-citations [ZZZ25] and [IJZ25] share authors with the present paper, but their roles are technical (pullback invariance of orbifold Chern classes, Hermite-Einstein metrics, definitions) and are not a uniqueness theorem or ansatz that forces the conclusion; the non-Kahler extension rests on the independent external inputs [Ou25a] and [CP25]. The skeptical counterexample to Corollary 4.13(2) (e.g. O(1) oplus O(-1) on P^1 x P^1) identifies a genuine non-sequitur -- 'E is alpha_1 * Omega-semistable by Remark 3.6' -- but that is a mathematical correctness gap, not a circular reduction: the failed implication does not make the target inequality identical to an assumption. The paper even flags related limitations (e.g. Section 5.1 on non-pluripolar products). No load-bearing step reduces by construction to its own input, so the circularity score is low.

Axiom & Free-Parameter Ledger

0 free parameters · 10 axioms · 0 invented entities

No fitted numerical constants and no new postulates. The central claim rests on a tower of cited theorems and domain assumptions, several of which come from recent preprints with overlapping authorship. The proof contributes the combinatorial and Hodge-index argument that combines these inputs, but it does not reduce the number of axioms a reader must accept.

axioms (10)
  • domain assumption Orbifold Chern classes of reflexive sheaves are well-defined and independent of the orbifold modification ([Ou25b, Lemma 2.2], [Ou24], [KO25]).
    This is the foundation for \hat c_2; the paper cites preprints for independence and does not reprove it.
  • domain assumption Functorial resolution of orbifold charts to smooth charts with group action exists ([DO23, Theorem 3.10]).
    Used in Definition 3.1 to obtain locally free orbi-bundles from torsion-free sheaves.
  • domain assumption Fujiki-class varieties admit bimeromorphic Kähler resolutions compatible with slope intersections.
    Used throughout Section 4 to transfer arguments from Kähler resolutions to X via equation (2.2).
  • domain assumption Harder-Narasimhan and Jordan-Hölder filtrations exist for torsion-free sheaves on Fujiki-class spaces with respect to products of nef classes (Setup 4.2).
    The paper asserts these follow 'by the same argument' as in [GKP16, DO23, ZZZ25] but does not give the proof in the non-Kähler singular setting.
  • domain assumption Positive-slope saturated subsheaves of tangent sheaves are algebraic foliations with rationally connected leaves ([Ou25a, Theorem 1.4], [CP25, Corollary 1.5]).
    This is the engine behind Proposition 4.3-4.5 and carries the main non-Kähler extension.
  • domain assumption The relative canonical bundle positivity lemma (Lemma 4.6) holds as in [CP25, Theorem 3.1].
    Lemma 4.6 is stated but its proof is only sketched by reference to [CP25]; it is load-bearing for Proposition 4.5.
  • standard math Orbifold Grothendieck-Riemann-Roch for embeddings holds ([MTTW25, Theorem 1.1]).
    Used in Proposition 3.10 to compute the second Chern class of torsion sheaves supported in codimension two.
  • domain assumption Hermite-Einstein metrics exist for stable Higgs orbibundles ([ZZZ25, Theorem 3.3], [CW24]).
    This is the analytic input for the orbifold Bogomolov-Gieseker inequality in Lemma 4.8; [ZZZ25] has overlapping authorship.
  • standard math Miyaoka's generic nefness and its Kähler generalizations hold ([Miy87], [Eno88], [Cao13], [Gue16]).
    Theorem 4.1 is imported and used to get the contradiction in Theorem 1.2.
  • standard math klt singularities imply quotient singularities in codimension two ([GK20, Lemma 5.8]).
    This justifies defining orbifold Chern classes for klt varieties, as stated in Remark 3.3.

pith-pipeline@v1.3.0-alltime-deepseek · 24515 in / 22547 out tokens · 192862 ms · 2026-08-01T14:36:40.566289+00:00 · methodology

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read the original abstract

We study inequalities for orbifold second Chern classes of compact normal analytic varieties in Fujiki's class. We prove Miyaoka's inequality for singular varieties in Fujiki's class with nef canonical divisor, as well as the semipositivity of the orbifold second Chern class for varieties with nef anti-canonical divisor. To prove these results, we establish generic nefness theorems for tangent and cotangent sheaves and an orbifold Bogomolov--Gieseker inequality for mixed polarizations.

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