REVIEW 3 major objections 3 minor 2 cited by
The Miyaoka-Yau inequality for singular varieties with big canonical or anticanonical divisors
T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read This paper proves the Miyaoka–Yau inequality for singular projective varieties whose canonical or anticanonical divisor is merely big, by pairing orbifold Chern classes with non-pluripolar products.
desk verdict The paper closes the nef-to-big gap for Miyaoka–Yau on klt varieties with genuinely new machinery, but the proof leans heavily on a couple of external stability inputs that the referee should verify. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing mechanism is the non-pluripolar product $\langle\alpha_1\cdots\alpha_p\rangle$, defined by pulling back to a resolution, multiplying the pulled-back currents with their pluripolar parts removed, and pushing forward. Together with orbifold Chern classes $\widehat{c}_i$, introduced via an orbifold modification $q:Z\to X$, this product gives an intersection number between the second Chern class and the movable class $\langle\alpha^{n-2}\rangle$. On the orbifold, Demailly's approximation theorem and the known Bogomolov–Gieseker inequality for Kähler orbifolds apply, yielding the Bogomolov–Gieseker inequality for $\langle\alpha^{n-1}\rangle$-semistable Higgs sheaves; the Miyaoka–Yau inequality follows from the stability of $\Omega^{[1]}_X\oplus\mathcal{O}_X$.
What would settle it
Take a projective klt threefold with big $K_X$ and compute the Harder–Narasimhan filtration of $\Omega^{[1]}_X$ with slope given by $\langle c_1(K_X)^2\rangle$; finding any destabilizing subsheaf disproves the semistability premise on which Theorem 1.1 rests.
Extended reading notes
Core claim
The central claim is that the Miyaoka–Yau inequality $(2(n+1)\widehat{c}_2(X)-n\widehat{c}_1(X)^2)\cdot\langle c_1(K_X)^{n-2}\rangle\ge0$ remains true for $n$-dimensional projective klt varieties with big $K_X$, and analogously with $-K_X$ for $K$-semistable klt varieties. The proof's engine is a Bogomolov–Gieseker inequality: any rank $r$ reflexive Higgs sheaf that is semistable with respect to the movable class $\langle\alpha^{n-1}\rangle$ of a big class $\alpha$ must satisfy $(2r\widehat{c}_2(E)-(r-1)\widehat{c}_1(E)^2)\cdot\langle\alpha^{n-2}\rangle\ge0$. Applying this to the Higgs sheaf $E=\Omega^{[1]}_X\oplus\mathcal{O}_X$ with Higgs field $(a,b)\mapsto(0,a)$, whose stability follows from the semistability of $\Omega^{[1]}_X$, yields the inequality. The formulation via non-pluripolar products is essential because big classes can have negative top self-intersections; the product $\langle\alpha^{n-2}\rangle$ discards the divisorial negative part and lands in homology, so the paper defines the needed intersection numbers through an orbifold modification.
Load-bearing premise
The proof relies on an imported theorem, not proved in this paper, that the reflexive cotangent sheaf of a canonical model is semistable with respect to the relevant big class; if that theorem is false, the chain of proof collapses.
Editorial extensions
If this is right
- Projective klt varieties with big canonical divisor satisfy the Miyaoka–Yau inequality in the non-pluripolar form, giving new Chern-number constraints in the big case.
- $K$-semistable projective klt varieties with big anticanonical divisor satisfy the same inequality with $c_1(-K_X)$, linking $K$-stability to Chern-class bounds.
- Miyaoka's inequality holds for non-uniruled or canonical-singularity varieties with nef $K_X$, and $\widehat{c}_2(X)\cdot\alpha_1\cdots\alpha_{n-2}\ge0$ holds when $-K_X$ is nef, even outside the Kähler category.
- The Bogomolov–Gieseker inequality for $\langle\alpha^{n-1}\rangle$-semistable Higgs sheaves on Moishezon quotient-singularity varieties is a new tool for stability questions in this singular setting.
- When $K_X$ is nef, the non-pluripolar product reduces to the ordinary intersection product, so the new theorem recovers previously known Miyaoka–Yau inequalities.
Reading between the lines
- If the paper's stability premise survives scrutiny, the same method should give Miyaoka–Yau inequalities for all compact varieties in Fujiki's class with quotient singularities in codimension two and rational singularities, not only projective ones.
- The inequality for $K$-semistable Fano-type varieties suggests a Chern-number obstruction to $K$-semistability: any variety violating the inequality cannot admit a $K$-semistable model with big anticanonical class.
- A natural testable extension is to check whether the vanishing property of big classes, which the paper verifies only in Moishezon, surface, and nef cases, holds for all compact Kähler varieties; a counterexample would remove a hypothesis from the main theorem.
- One could probe the equality case: if equality holds in the big-canonical inequality, the structure of $X$ (torus fibrations, ball quotients, and so on) might be recovered, as is known in the nef case.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper establishes Miyaoka–Yau type inequalities for singular compact complex analytic varieties in the case where the canonical or anticanonical divisor is big rather than nef. The main results are Theorem 1.1, which asserts the inequality (2(n+1)bc2(X)−nbc1(X)^2)·⟨c1(K_X)^{n-2}⟩≥0 for projective klt varieties with big K_X and for a class of varieties with quotient singularities in codimension two and rational singularities, and Theorem 1.2, the analogous inequality when −K_X is big and the variety is K-semistable. The proof proceeds by defining non-pluripolar products with respect to big classes on singular varieties, developing an orbifold intersection calculus for orbifold Chern classes against these products, and proving a Bogomolov–Gieseker inequality for ⟨α^{n−1}⟩-semistable reflexive Higgs sheaves. The paper also proves Miyaoka-type inequalities and semipositivity of the second Chern class in nef settings, and discusses equality cases.
Significance. If the proof is completed, the main results constitute a substantial extension of the classical Miyaoka–Yau inequality: the big case is genuinely new, and the formulation through non-pluripolar products is natural and well-motivated. The paper also contains useful structural contributions, including a definition of non-pluripolar products for singular varieties in Fujiki’s class, an orbifold reduction for the second Chern class, and a Bogomolov–Gieseker inequality for Higgs sheaves with respect to big classes. The authors are explicit about many delicate technical points, such as the vanishing property for non-pluripolar products and the role of rational singularities in pullbacks. The central caveat is that load-bearing stability and orbifold Riemann–Roch inputs are imported from preprints, some by the same authors, and at least one of these inputs is explicitly flagged in the text as not yet established in the needed form.
major comments (3)
- [Section 5.1, first paragraph] The proof of Theorem 1.1 begins with the assertion that Ω^[1]_X is ⟨c_1(K_X)^{n−1}⟩-semistable, citing [Jin25, Example 4.11] and ultimately [Gue16, Theorem A]. This assertion is load-bearing: the whole proof depends on the Higgs sheaf E = Ω^[1]_X ⊕ O_X being ⟨c_1(K_X)^{n−1}⟩-stable, and that stability of E rests on semistability of its first factor. The present paper does not prove this semistability statement, and the transfer from X_can to X uses [Jin25, Theorem 5.1], whose hypotheses include an α-negative bimeromorphic contraction and the vanishing property. The text only says 'since f is a c_1(K_X)-negative bimeromorphic contraction' and does not verify the precise hypotheses or quote the exact form of [Gue16, Theorem A]. This creates a concrete correctness risk: for instance, if [Gue16, Theorem A] concerns the tangent sheaf under nef −K_X, or if the K_X-MMP map is not shown to satisfy all hypotheses of [Jin25, Theorem 5.1], the construction of the stable Higgs bundle collapses. The authors should either prove the needed semistability statement directly or state and verify the exact external results on which it depends.
- [Remark 4.24] Remark 4.24 explicitly acknowledges that a stronger orbifold Grothendieck–Riemann–Roch statement has not yet been established. In the proof of Proposition 4.23, however, a formula of the same shape is used for a torsion orbi-sheaf supported in codimension two, namely c^orb_2(Q_orb) = −Σ a_l {S_l,orb} with a_l ≥ 0, justified by citing Theorem 4.21(3). The paper should clarify exactly what is proved by [MTTW25] in the Bott–Chern setting and what is only conjectural, and should state whether the main theorems of the paper depend on the unresolved stronger statement. At present, Proposition 4.23, and through it Lemma 6.10 and the semistable reduction step in Theorem 4.30, relies on a circle of orbifold Riemann–Roch facts whose available scope is not made precise.
- [Proposition 4.16 and §4.2.1] The definition of the intersection number bc_2(E)·⟨α_1⋯α_{n−2}⟩ is foundational for the orbifold reduction in the proof of Theorem 1.8 and for the statement of Theorems 1.1 and 1.2. The proof of Proposition 4.16 is a local-chart sketch that refers to [Bla96, Lemma 1.10] and uses a diagram of bimeromorphic maps without giving all compatibilities among orbifold structures, ramification degrees, and non-pluripolar products. Since the non-pluripolar product is defined by pushforward from a resolution rather than by a bimeromorphically invariant class in H^*(X,R), the independence proof is more delicate than the text indicates. A fully detailed verification is needed before the definition can be considered established.
minor comments (3)
- [Section 6.3, proof of Lemma 6.10] In equation (6.6), the term bc_1(G_i)^2 is written with G_i rather than G_i^{∨∨}; since G_i is only torsion-free, the orbifold first Chern class is defined for its double dual, consistent with the surrounding formulas.
- [Abstract and Theorem 4.30] The abstract states the Bogomolov–Gieseker inequality for sheaves semistable with respect to a big class α, but Theorem 4.30 assumes the vanishing property for α. The abstract and introduction should state this hypothesis explicitly, since it is essential and is only automatic in the Moishezon, two-dimensional, or nef cases.
- [Section 5.2] In the proof of Theorem 1.2, the statement that the canonical extension sheaf E_X is ⟨c_1(−K_X)^{n−1}⟩-semistable repeats the same pattern as in Theorem 1.1: the stability of E_Z on the K-semistable Fano model is imported from [DGP24, Remark 2.4 and Theorem 3.3], and the transfer through Theorem 5.1 is only summarized. The relevant hypotheses should be verified explicitly here as well.
Circularity Check
No significant circularity: the main inequality is derived from external semistability inputs and an independently proven Bogomolov-Gieseker inequality; reliance on author preprints [Jin25] and [ZZZ25] is load-bearing but not circular.
full rationale
I traced the proof of Theorem 1.1. The argument reduces the Miyaoka-Yau inequality to the Bogomolov-Gieseker inequality (Theorem 4.30) applied to the reflexive Higgs sheaf E = Omega^[1]_X direct-sum O_X. The needed semistability of Omega^[1]_X is imported rather than proved from the target inequality: the paper states 'According to [Gue16, Theorem A], the reflexive cotangent sheaf Omega^[1]_Xcan is c1(K_Xcan)^{n-1}-semistable' and then transfers this to X via [Jin25, Theorem 4.9]. This is an external input, and [Gue16] is a published source independent of the present authors; the transfer statement from [Jin25] is a general stability-comparison result, not the Miyaoka-Yau inequality. The Bogomolov-Gieseker inequality itself is proved in Section 4 using orbifold BG inputs from [ZZZ25, Cor 1.3] and [Ou24], again prior-work inputs rather than assumptions of Theorem 1.1. I found no place where the claimed inequality is inserted as a definition, where a fitted parameter is renamed as a prediction, or where a self-citation alone carries the entire argument without independent mathematical content. The self-referential limitations and open questions in the paper were also checked; they concern equality cases and psef extensions, not the derivation chain of Theorems 1.1-1.5. Therefore the derivation is not circular; the score 2 reflects the noticeable but non-circular reliance on coauthor preprints for load-bearing stability statements.
Assumptions & free parameters
assumptions (3)
- domain assumption Vanishing property (Definition 3.23) for big classes
- domain assumption Correctness of external preprints [Jin25], [ZZZ25], [Ou24], [Xu23], [DGP24]
- standard math Klt singularities have quotient singularities in codimension two (cited to [GK20, Lemma 5.8])
Cite this review
Pith. "Pith review of The Miyaoka-Yau inequality for singular varieties with big canonical or anticanonical divisors." pith.science (2026). https://pith.science/paper/ZFB6CYN5
@misc{pith2026250708522,
author = {Pith},
title = {Pith review of: The Miyaoka-Yau inequality for singular varieties with big canonical or anticanonical divisors},
year = {2026},
howpublished = {\url{https://pith.science/paper/ZFB6CYN5}},
note = {Machine review of arXiv:2507.08522}
}
abstract
We establish the Miyaoka-Yau inequality for $n$-dimensional projective klt varieties with big canonical divisor $K_X$: \[ (2(n+1)\widehat{c}_2(X) - n \widehat{c}_1(X)^2) \cdot \langle c_1(K_X)^{n-2} \rangle \ge 0. \] We also prove the Miyaoka-Yau inequality for K-semistable projective klt varieties with big anticanonical divisor $-K_X$. As part of our approach, we define the non-pluripolar product $\langle \alpha_1 \cdots \alpha_p \rangle$ on singular varieties, and establish the Bogomolov-Gieseker type inequality for $\langle \alpha^{n-1} \rangle$-semistable Higgs sheaves with respect to a big class $\alpha$.
Forward citations
Cited by 2 Pith papers
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The Miyaoka-Yau inequality and the delta invariant for Fano varieties
Every klt Fano variety satisfies a Miyaoka–Yau inequality whose deficit is controlled by (1−min{1,δ(X)})², and every Fano manifold with a Kähler–Ricci soliton satisfies the analogous equivariant inequality.
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Semipositivity of the orbifold second Chern class in Fujiki's class
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 ...
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