{"id":"ebf036e3-1408-4bb6-aa8b-72aef1be2072","arxiv_id":"2507.08522","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For projective klt varieties with big canonical or anticanonical divisor, the Miyaoka-Yau Chern class inequality holds when intersections are taken with the non-pluripolar product.","lead":"This paper proves the Miyaoka-Yau inequality for singular algebraic varieties whose canonical (or anticanonical) divisor is big, using a new non-pluripolar intersection product. It extends a classical Chern-class inequality from smooth manifolds to klt singular spaces, with applications to the minimal model program.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.1 hinges on an unverified external stability statement for Ω^[1] on canonical models ([Gue16, Thm A] via [Jin25, Thm 5.1]); if that input is wrong or inapplicable, the Higgs bundle construction collapses.","rationale":"The strongest claim is Theorem 1.1. The internal architecture—non-pluripolar products on singular spaces, orbifold reduction, Bogomolov–Gieseker for ⟨α^{n-1}⟩-semistable Higgs sheaves, and application to Ω^1⊕O—is coherent, and the paper gives real proofs for the new analytic machinery. The BG inequality (Theorem 4.30) is derived in the text with only standard external tools (Demailly approximation, [ZZZ25] orbifold BG). The load-bearing step is the semistability of the cotangent sheaf on the canonical model, imported from [Gue16, Theorem A] and transferred by [Jin25, Theorem 5.1]. Both are recent preprints by the second author or close collaborators, and neither is reproduced. The risk is not that the theorem is false, but that a hidden mismatch of hypotheses would void the first step of the proof. This is precisely the reader's weakest assumption, and I agree. The verdict should remain conditional: the paper is promising but not fully verified until the external stability input is confirmed or incorporated.","tokens_in":58662,"tokens_out":19226,"duration_ms":228585,"concrete_test":"Verify the external input directly: (1) obtain the exact statement of [Gue16, Theorem A] and check that it indeed proves semistability of Ω^[1]_{Xcan} with respect to the class c1(K_Xcan)^{n-1} for a projective klt canonical model (ample K_Xcan), and not merely semistability of T_X for nef -K_X; (2) check that [Jin25, Theorem 5.1] applies to the birational contraction f:X→Xcan for α=c1(K_X), including the existence of the effective q-exceptional divisor E and the Moishezon vanishing property; (3) if both statements check out, the concern is resolved; if not, the proof of Theorem 1.1 needs a replacement stability argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the first paragraph of §5.1, the paper asserts: 'According to [Gue16, Theorem A], the reflexive cotangent sheaf Ω^[1]_{Xcan} is c1(K_Xcan)^{n-1}-semistable', and then transfers this to X using [Jin25, Theorem 5.1]. This is not a peripheral citation: the entire proof of Theorem 1.1 depends on E = Ω^[1]_X ⊕ O_X being ⟨c1(K_X)^{n-1}⟩-stable as a Higgs sheaf, and stability of the first factor is precisely the imported semistability. The manuscript does not prove or reproduce either external result. A concrete risk is a mismatch of hypotheses: [Gue16, Theorem A] may treat the tangent sheaf under nef -K_X rather than the cotangent sheaf under nef/ample K_X; and [Jin25, Theorem 5.1] requires f to be an α_X-negative bimeromorphic contraction with the vanishing property, and the verification that the K_X-MMP map X→Xcan satisfies this is only summarized ('since f is a c1(K_X)-negative bimeromorphic contraction'). If either external premise is misstated or requires an additional hypothesis (e.g., K_X nef rather than merely big on the model), Theorem 1.1 fails at its first step, and the Miyaoka–Yau inequality in the big case is unsupported. The reader's conditional verdict is therefore appropriate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":58996,"tokens_out":6412,"duration_ms":79285,"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":[{"comment":"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.","section":"Section 5.1, first paragraph"},{"comment":"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.","section":"Remark 4.24"},{"comment":"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.","section":"Proposition 4.16 and §4.2.1"}],"minor_comments":[{"comment":"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.","section":"Section 6.3, proof of Lemma 6.10"},{"comment":"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":"Abstract and Theorem 4.30"},{"comment":"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.","section":"Section 5.2"}],"recommendation":"major_revision","confidential_remarks":"The paper depends heavily on preprints, including [Jin25], [IMM24], and [ZZZ25], several of which are by the same authors or overlapping author groups. Some of these inputs are not yet independently refereed, and the manuscript explicitly flags an orbifold Grothendieck–Riemann–Roch statement as missing. I would recommend that the editor seek verification of the external stability and orbifold Riemann–Roch inputs before publication, even if the final version only cites them in a more transparent and precise way."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorems are new and significant: the Miyaoka–Yau inequality for projective klt varieties with big K_X, and for K-semistable klt varieties with big -K_X, formulated via non-pluripolar products. That is the right statement for big classes, since K_X^{n-2} can be negative. The paper also defines non-pluripolar products on singular varieties and proves a Bogomolov–Gieseker inequality for Higgs sheaves semistable with respect to a big class. That is a substantial toolkit, and the exposition is careful. The authors are honest about overlaps: they flag Corollary 1.6 as already in [ZZZ25] and Theorem 1.5 as independently obtained in [MWWZ25].\n\nThe soft spots are real but not fatal. The proof of Theorem 1.1 depends on the assertion from [Gue16, Theorem A] that Ω^[1]_{Xcan} is c1(K_Xcan)^{n-1}-semistable, transferred to X via [Jin25, Theorem 5.1]. That is a load-bearing external premise, and [Jin25] is a preprint by the second author. The verification that the MMP map X→Xcan is a c1(K_X)-negative bimeromorphic contraction is summarized in a couple of lines. If that transfer or the cited theorem has an extra hypothesis (e.g., K_X nef rather than big), the construction of the stable Higgs sheaf collapses. A referee needs to check that step directly. Similarly, Theorem 1.2 relies on [Xu23] for finite generation of the anticanonical ring; that is a deep external result, though a published one. Remark 4.24 explicitly notes the missing orbifold Grothendieck–Riemann–Roch; that does not affect the main inequality, but it limits the equality analysis.\n\nOn balance the central argument looks credible. The dependencies are heavy but they are concentrated and identifiable. This is not a circular argument: the non-pluripolar product and orbifold BG inequality are used to prove MY, not assumed. I think the paper deserves a serious referee. The conditional verdict is right: accept only after the external stability input is confirmed, or the authors include a proof.\n\nRecommendation: send to a good journal, ask the referee to verify [Gue16]'s theorem applies to the cotangent sheaf on the canonical model with the non-pluripolar polarization, and to check the contraction assertion in Section 5.1. If that step holds, the paper is a strong contribution.","headline":"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.","tokens_in":59583,"tokens_out":3848,"would_cite":true,"duration_ms":42841,"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 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.","keywords":["Miyaoka-Yau inequality","Bogomolov-Gieseker inequality","non-pluripolar product","orbifold Chern classes","Higgs sheaves","klt singularities","K-semistability","big canonical divisor"],"falsifier":"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.","tokens_in":58466,"feed_emoji":"📐","tokens_out":7598,"duration_ms":73263,"temperature":0.7,"pith_summary":"This paper tries to establish the Miyaoka–Yau inequality—a lower bound on a combination of Chern classes—for singular projective varieties whose canonical or anticanonical divisor is big but not necessarily ample. The classical inequality is known when these divisors are ample or nef; the new point is to allow merely big classes by pairing the orbifold Chern classes with a non-pluripolar product, which ignores the negative part of a big class. The authors prove the inequality for $n$-dimensional projective klt varieties with big canonical divisor, and for $K$-semistable projective klt varieties with big anticanonical divisor. Along the way they define non-pluripolar products on singular varieties and prove a Bogomolov–Gieseker inequality for semistable Higgs sheaves in this setting. A sympathetic reader would care because this gives Chern-number constraints for a much wider class of singular varieties than previously known.","feed_headline":"Big canonical divisors keep Miyaoka–Yau inequality true","feed_subtitle":"Non-pluripolar products let orbifold Chern classes see the positive part of a big divisor.","key_machinery":"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 $\\Omega^{[1]}_X\\oplus\\mathcal{O}_X$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the semistability criterion for sheaves with big classes and the stability of $\\Omega^{[1]}_X$ used in the proof of Theorem 1.1.","marker":"[Jin25]"},{"why":"Proves semistability of the cotangent sheaf on canonical models, the external anchor for the stability premise.","marker":"[Gue16, Theorem A]"},{"why":"Introduces non-pluripolar products on compact Kähler manifolds, which the paper extends to singular varieties.","marker":"[BEGZ10]"},{"why":"Provides the orbifold modification used to define intersection numbers with $\\widehat{c}_2$.","marker":"[Ou24]"},{"why":"Extends Demailly's approximation theorem to orbifolds, needed in the Bogomolov–Gieseker proof.","marker":"[Wu23b]"},{"why":"Supplies the orbifold Bogomolov–Gieseker inequality and the Miyaoka–Yau inequality for minimal klt spaces used as input.","marker":"[ZZZ25]"},{"why":"Gives the existence of canonical models for klt varieties of log general type.","marker":"[BCHM10]"},{"why":"Gives finite generation of the anticanonical ring and the Proj construction for $K$-semistable varieties with big anticanonical class.","marker":"[Xu23]"},{"why":"Proves semistability of the canonical extension sheaf for $K$-semistable Fano varieties, used in Theorem 1.2.","marker":"[DGP24]"},{"why":"Provides the orbifold Chern character needed for the exact-sequence estimates in Miyaoka's inequality.","marker":"[MTTW25]"}],"fun_headline_variants":["Miyaoka-Yau inequality now extends to singular klt varieties","Big divisors unlock Miyaoka-Yau for singular varieties","Bogomolov-Gieseker inequality powers new Miyaoka-Yau proof","Non-pluripolar products generalize Miyaoka-Yau inequality","Miyaoka-Yau holds for big canonical divisors on klt spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Miyaoka-Yau inequality now extends to singular klt varieties","Big divisors unlock Miyaoka-Yau for singular varieties","Bogomolov-Gieseker inequality powers new Miyaoka-Yau proof","Non-pluripolar products generalize Miyaoka-Yau inequality","Miyaoka-Yau holds for big canonical divisors on klt spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00045,"raw_usage":{"total_tokens":2287,"prompt_tokens":986,"completion_tokens":1301,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":1208}},"tokens_in":602,"tokens_out":1301,"duration_ms":12609,"temperature":1.0,"reasoning_tokens":1208,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:16:12.225850+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}