{"id":"fe538c1e-c595-4ba2-936d-9569166f6776","arxiv_id":"2607.25181","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper proves a Chern-number inequality for all Fano varieties, a central class of curved spaces in algebraic geometry, with the allowed violation of the classical Miyaoka–Yau bound controlled by the delta invariant, a modern stability measure. The same inequality is extended to manifolds carrying Kähler–Ricci solitons, using a new equivariant intersection theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorems hinge on external slope estimates (Theorem 7.1 / Theorem 6.2) whose proofs are not fully verified here; the appendix's alternative proof omits the key analytic estimates (Claim A.6, Terms (I)/(III)), so the central inequality is only as secure as those inputs.","rationale":"The reader's weakest-assumption analysis matches my read: the theorems are assembled from external slope estimates, and the paper's own proof of the klt input omits precisely the hard analytic estimates. The internal algebra from the slope bounds to the Miyaoka–Yau inequalities is straightforward and appears correct, and the sharpness examples are consistent. The unresolved points—Term (I)/(III) in Claim A.6, the omitted Corollary 5.2 proof, and the Remark 3.10 semistability mismatch—are not demonstrated errors, but they are concrete gaps in the verification of the central claims. Because the main theorems would collapse if the quoted slope estimates failed, a conditional verdict is appropriate. That is already the reader's verdict, so no adjustment is needed.","tokens_in":57796,"tokens_out":14062,"duration_ms":130052,"concrete_test":"Complete the omitted estimates for Term (I) and Term (III) in Claim A.6 (Appendix A) by following the cited 'same as in [DGP24]' arguments and verifying that lim_{t,ε→0} a_{t,ε} ≤ (1−γ)c_1(X)^n holds in the singular klt setting. If these limits do not vanish, Theorem 7.1 lacks the self-contained proof claimed; alternatively, obtain and independently check the full proof of [XZ26, Lemma 5.9], and test its slope bound on the weighted projective space P(1,1,1,2) by comparing with the sharpness computation in Example 7.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing point is not the short Chern-number composition after the slope estimates are accepted; it is the klt Tian slope estimate itself. Theorem 1.1 depends on Theorem 7.1, quoted from [XZ26, Lemma 5.9], which supplies the two-sided bounds on µmax/µmin of the canonical extension sheaf. The paper's own alternative proof in Appendix A explicitly omits the two analytic estimates needed: Claim A.6 asserts that Term (I) and Term (III) vanish in the ε,t→0 limits, and then says 'we therefore omit the proof' (§A.0.8). These terms control the singular contributions of the twisted Kähler–Einstein metrics on a resolution, and without them the asserted bound (A.3) is not established.\n\nThe equivariant Theorem 1.2 has the same architecture: it imports Theorem 6.2 from [HL26, Theorems 7.1/7.2] and uses Corollary 5.2, whose proof is also omitted. Remark 3.10 explicitly leaves open whether the paper's v-semistability agrees with [HL26]'s semistability; if the imported theorem relies on a different stability notion, the transfer could break. No internal error is identified in the short derivation from these inputs; the central claims are only as secure as the unverified external estimates.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes a Miyaoka-Yau-type inequality for arbitrary n-dimensional projective klt Fano varieties, with a defect controlled by the delta invariant: (2(n+1)c^_2 - n c_1^2).c_1^{n-2} >= -n(1-min{1,delta})^2.c_1^n. The inequality is sharp, with equality on weighted projective spaces P(1,...,1,b). It also develops an equivariant version for smooth Fano G-manifolds, replacing Chern classes and delta invariant by equivariant Chern classes and weighted delta invariant, and derives as a corollary the equivariant Miyaoka-Yau inequality for every smooth Fano manifold admitting a Kahler-Ricci soliton. The proof strategy follows Tian's canonical extension sheaf: it combines a slope estimate for this sheaf (quoted from [XZ26] in the klt case and from [HL26] in the soliton setting) with a suitable Langer inequality. The internal composition from these slope estimates to the Chern-number inequalities is short and checkable. Examples and computations are supplied, including an explicit verification for the blow-up of CP^2 at a point.","tokens_in":1647,"tokens_out":2636,"duration_ms":81353,"significance":"If the external slope estimates are correct, the paper gives a substantial unification: when min{1,delta(X)}=1 it recovers the known Miyaoka-Yau inequality for K-semistable Fano varieties, while for K-unstable Fanos it provides a sharp quantitative defect. The equivariant extension to Kahler-Ricci solitons is new and naturally complements recent work of Hallam-Lahdili and Inoue. The paper is honest about its dependencies, and the numerical examples are concrete. The main weakness is that the load-bearing slope estimates are not fully proved in the manuscript: Theorem 7.1 is quoted from a contemporaneous preprint, its alternative proof in Appendix A explicitly omits key analytic estimates, and the equivariant part may rely on a different stability notion. These issues are local to the imports but central for the claimed theorems.","major_comments":[{"comment":"The proof of Theorem 1.1 rests on Theorem 7.1, quoted from [XZ26, Lemma 5.9], a contemporaneous preprint not verified in this paper. The alternative proof in Appendix A is incomplete: Claim A.6 asserts limits for Term (I) and Term (III), but §A.0.8 says 'The estimates of Term (I) and Term (III) are the same as in [DGP24] ... We therefore omit the proof.' These terms control the singular contributions on the resolution, and without them inequality (A.3) is not established. Since Theorem 1.1 is central, readers cannot verify the main theorem from the manuscript alone. The authors should either supply a complete proof of Claim A.6 or state Theorem 1.1 as explicitly conditional on [XZ26] with a precise statement of the imported result.","section":"§7.1, Theorem 7.1; Appendix A.0.8, Claim A.6"},{"comment":"Theorem 1.2 depends on Theorem 6.2, quoted from [HL26, Theorems 7.1/7.2]. However, Remark 3.10 explicitly states that the paper does not know whether its v-semistability (Definition 3.9) agrees with [HL26, Definition 5.3]. Since Theorem 6.2 is stated in [HL26]'s framework, applying it to the present setup requires a comparison lemma or a proof of Theorem 6.2. Without this, the equivariant Miyaoka-Yau inequality may not follow from the quoted result. This is load-bearing for Theorem 1.2 and Corollary 1.3.","section":"§6.1, Theorem 6.2; Remark 3.10"},{"comment":"Corollary 5.2, the Bogomolov-Gieseker inequality for exponential weights, is asserted without proof ('...hence we omit the proof'). The exponential weight is obtained as a limit of polynomial weights, and v-semistability is not obviously preserved under this approximation. Semistability is not an open condition in the weight in any evident way, so the limiting argument requires justification. This inequality is used in Theorem 5.3(2) for the equivariant Langer inequality and hence feeds into Theorem 1.2. The omitted proof should be supplied.","section":"§5.2, Corollary 5.2"}],"minor_comments":[{"comment":"The notation for the orbifold second Chern class is inconsistent: the abstract uses hat c_2, while the main text often uses bc_2. Please unify the notation and define it once in the introduction.","section":"§1 and §7"},{"comment":"The value delta(X)=500/767 is quoted from [ZZ22, Theorem 1.1]. Since this example is used to illustrate that the Miyaoka-Yau inequality fails while the delta bound holds, a short indication of the computation or a more precise reference to the formula would help the reader verify the arithmetic.","section":"Example 7.4"},{"comment":"The proof of Claim B.5 uses concavity of g(lambda)=f(lambda)^(1/(n-1)) from the Brunn-Minkowski inequality; this is stated without a reference. The appendix is speculative and not load-bearing, but a reference would improve readability.","section":"Appendix B, Claim B.5"}],"recommendation":"major_revision","confidential_remarks":"This is a promising paper with a clear and mostly checkable internal structure, but the main theorems are currently conditional on slope estimates whose proofs are not fully contained in the manuscript. The authors are honest about the dependencies, including the omitted estimates in Claim A.6 and the stability-notion ambiguity in Remark 3.10. I would support acceptance if the authors either provide complete proofs of these inputs or restate the main results as theorems conditional on explicitly formulated external results. The equivariant part also needs the stability-notion comparison resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. The main theorem is real news: a Miyaoka–Yau inequality for every klt Fano variety, K-stable or not, with the defect measured by (1−δ'(X))^2, sharp for weighted projective spaces. I don't see that statement in the cited literature, and it lands exactly where you'd expect — the K-semistable case (δ'=1) recovers the known results. Second, the proof is a short composition: the klt Tian slope estimate from [XZ26, Lemma 5.9] plus the klt Langer inequality from [IMM25, Prop. 3.4]. I traced the composition in Section 7; it is clean and checkable. The trouble is that the deepest input, the klt slope estimate, is imported from a contemporaneous 2026 preprint and not verified here.\n\nWhat the paper does well: Sections 2–5 build an algebraic weighted Bogomolov–Gieseker/Langer machinery for torsion-free sheaves via Edidin–Graham equivariant intersection theory. That is substantial, mostly self-contained work — the semi-simple fibration construction in Section 4 has real content — and it yields the equivariant Theorem 1.2 and Corollary 1.3 for Kähler–Ricci solitons, which are genuinely new. The sharpness computation for P(1,…,1,b) is direct and consistent.\n\nSoft spots, in proportion. The appendix's alternative proof of the klt slope estimate omits exactly the two analytic estimates that control the singular contributions: Claim A.6, Terms (I) and (III), are dismissed with \"we therefore omit the proof.\" So the paper's own proof attempt stops at the hard part. The equivariant chain has the same architecture: Theorem 6.2 is quoted from [HL26], the proof of Corollary 5.2 is skipped, and Remark 3.10 explicitly says the authors don't know whether their v-semistability agrees with [HL26]'s — if the imported theorem depends on that, the transfer could break. None of these is a demonstrated error; each is a concrete item for the authors to fix or point to. The appendix on Donaldson's conjecture is honestly conditional and gives a weaker constant, so it doesn't threaten anything.\n\nBottom line: anyone working on K-stability or Fano varieties will want this inequality, and it will be cited regardless. The paper deserves a serious referee, but the referee should send it back asking for the quoted inputs to be verified in detail, and for Claim A.6 to be completed or removed. I'd take it to reading group.","headline":"A genuinely new, sharp δ-controlled Miyaoka–Yau inequality with a clean short proof — once you accept the imported slope estimates, which are contemporaneous, unverified preprints.","tokens_in":58680,"tokens_out":3441,"would_cite":true,"duration_ms":28541,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J45","14C17","32Q20","32Q26"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every n-dimensional klt Fano variety, the Miyaoka–Yau Chern-number inequality holds up to a correction term controlled by the square of (1−δ), with equality on weighted projective spaces; the same mechanism works equivariantly for Kähle","keywords":["Miyaoka–Yau inequality","Fano varieties","delta invariant","K-stability","Kähler–Ricci solitons","equivariant Chern classes","weighted delta invariant","klt singularities"],"falsifier":"Compute the two omitted limits in the appendix (the asymptotic behaviour of Term (I) and Term (III) as ε,t→0): if either limit is non-zero, the appendix proof collapses and Theorem 1.1 depends entirely on the external slope estimate. Independently, one can test the theorem's output on the explicit toric Fano fourfold of Example 7.4, where both sides are computed exactly and the predicted inequality holds.","tokens_in":57547,"feed_emoji":"📐","tokens_out":10758,"duration_ms":95098,"temperature":0.7,"pith_summary":"The paper's central claim is that the Miyaoka–Yau inequality, classically known when a Fano variety is K-semistable or admits a Kähler–Einstein metric, holds universally in a corrected form governed by the delta invariant δ. For any n-dimensional projective klt Fano variety, (2(n+1)ĉ₂ − n c₁²)·c₁^{n−2} is bounded below by −n(1−min{1,δ})²·c₁ⁿ, so the only possible violation of the classical inequality is a defect quadratic in how far δ is from 1. The bound is sharp: weighted projective spaces P(1,…,1,b) attain equality while failing the classical inequality. The same proof, phrased with equivariant Chern classes and the weighted delta invariant, yields an equivariant version, and as a consequence the equivariant Miyaoka–Yau inequality holds for every smooth Fano manifold admitting a Kähler–Ricci soliton — in particular for toric Fano manifolds.","feed_headline":"Delta invariant controls the Miyaoka–Yau defect on all Fano varieties","feed_subtitle":"A sharp bound now covers K-unstable Fano varieties, and Kähler–Ricci solitons follow equivariantly.","key_machinery":"The load-bearing object is the canonical extension sheaf V, the reflexive sheaf in 0→O_X→V→T_X→0 whose extension class is a multiple of c₁(X) chosen from the delta invariant. The argument reduces the Chern-number inequality to two ingredients concerning V: a lower bound on its minimal slope (degree per unit rank) of the form µ_min(V) ≥ δ′(X)/(n+1)·c₁(X)ⁿ, and a discriminant inequality that bounds 2(n+1)c₂−n c₁² from below in terms of the spread between the maximal and minimal slopes. In the equivariant/soliton setting, slopes are taken with respect to the weight v(µ)=e^{⟨µ,ξ⟩}; algebraic equivariant intersection theory is used so that the second equivariant Chern class is defined for torsion","core_discovery":"On the paper's own terms, the discovery is that the Miyaoka–Yau inequality is controlled by the delta invariant rather than by K-semistability. For every n-dimensional projective klt Fano variety X, with ĉ₂(X) the orbifold second Chern class, (2(n+1)ĉ₂(X) − n c₁(X)²)·c₁(X)^{n−2} ≥ −n(1−δ′(X))²·c₁(X)ⁿ, where δ′(X)=min{1,δ(X)} and δ(X) is the delta invariant, equivalently the greatest Ricci lower bound. The inequality is sharp: P(1,…,1,b) gives equality while violating the un-corrected inequality. The equivariant counterpart replaces the usual Chern classes by torus-equivariant Chern classes and the delta invariant by the weighted delta invariant; in particular, a smooth Fano manifold admittin","pith_inferences":["The quadratic dependence on (1−δ′) is likely structural: any invariant that bounds the minimal slope of the canonical extension sheaf should enter the Chern-number defect through the same square, because the discriminant inequality is quadratic in the slope gap.","The equivariant framework makes the inequality computable from moment-polytope data; a toric classification of equality cases beyond P(1,…,1,b) is now a combinatorial question the paper leaves open.","If the same slope-lower-bound input is established for log Fano pairs or for other weights, the same two-step proof would produce Miyaoka–Yau-type inequalities in those settings without new geometric ideas.","A natural open question is whether equality in the soliton inequality characterizes equivariantly projectively flat manifolds or weighted projective spaces; the paper does not address this."],"forward_implications":["K-semistable Fano varieties satisfy the classical Miyaoka–Yau inequality, since δ′=1; the new theorem contains that known case.","The bound cannot be improved in general: the weighted projective spaces P(1,…,1,b) attain equality and at the same time fail the classical inequality.","Every smooth Fano manifold admitting a Kähler–Ricci soliton satisfies the equivariant Miyaoka–Yau inequality; in particular, all toric Fano manifolds do.","When the soliton is a Kähler–Einstein metric (ξ=0), the equivariant statement reduces to the usual Miyaoka–Yau inequality.","Equality in the non-equivariant theorem forces the canonical extension sheaf to be either slope-semistable or to have a Harder–Narasimhan filtration of length two whose destabilizing subsheaf has rank one."],"fun_headline_variants":["Delta invariant sharpens Miyaoka-Yau for all Fano varieties","Miyaoka-Yau bound now covers K-unstable Fano varieties","Equivariant Miyaoka-Yau inequality for Kähler-Ricci solitons","Sharp Miyaoka-Yau inequality via delta invariant"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire theorem rests on the quoted slope estimate for the canonical extension sheaf of a klt Fano variety, µ_min(V) ≥ δ′(X)/(n+1)·c₁(X)ⁿ, and the paper's own appendix proof of that estimate is explicitly incomplete because it omits the asymptotic bounds for two terms (Term (I) and Term (III)).","fun_headline_variants_meta":{"raw":{"variants":["Delta invariant sharpens Miyaoka-Yau for all Fano varieties","Miyaoka-Yau bound now covers K-unstable Fano varieties","Equivariant Miyaoka-Yau inequality for Kähler-Ricci solitons","Sharp Miyaoka-Yau inequality via delta invariant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000803,"raw_usage":{"total_tokens":3370,"prompt_tokens":754,"completion_tokens":2616,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":2536}},"tokens_in":498,"tokens_out":2616,"duration_ms":20401,"temperature":1.0,"reasoning_tokens":2536,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T03:15:00.090329+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the two omitted limits in the appendix (the asymptotic behaviour of Term (I) and Term (III) as ε,t→0): if either limit is non-zero, the appendix proof collapses and Theorem 1.1 depends entirely on the external slope estimate. Independently, one can test the theorem's output on the explicit toric Fano fourfold of Example 7.4, where both sides are computed exactly and the predicted inequality holds.","supporting_citations":[],"review_version":1}