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The Miyaoka-Yau inequality and the delta invariant for Fano varieties

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Pith's one-line read 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

desk verdict 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. read the letter →

arxiv 2607.25181 v1 pith:QKNCI4NL submitted 2026-07-28 math.AG math.CVmath.DG

Tomoyuki Hisamoto , Masataka Iwai This is my paper · ORCID
classification math.AGmath.CVmath.DG MSC 14J4514C1732Q2032Q26
keywords Miyaoka–YauinequalityFanovarietiesdeltainvariantK-stabilityKähler–RiccisolitonsequivariantChernclassesweightedkltsingularities
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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)).

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

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.

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 (3)
  1. [§7.1, Theorem 7.1; Appendix A.0.8, Claim A.6] 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.
  2. [§6.1, Theorem 6.2; Remark 3.10] 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.
  3. [§5.2, Corollary 5.2] 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.
minor comments (3)
  1. [§1 and §7] 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.
  2. [Example 7.4] 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.
  3. [Appendix B, Claim B.5] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Chern-number inequality is derived from external slope estimates and a Langer inequality; no equation consumes its own conclusion.

full rationale

The derivation chain is non-circular. Theorem 1.1 is obtained by feeding the klt Tian slope estimate (Theorem 7.1, quoted from [XZ26, Lemma 5.9]) into the klt Langer inequality [IMM25, Prop. 3.4]: the inequality (2(n+1)ĉ2(X)−n c1(X)^2)·c1(X)^{n−2} ≥ −n(1−δ'(X))^2·c1(X)^n is derived from bounds on µ_max, µ_min of the canonical extension sheaf. The delta invariant enters only as the slope bound's input; the output is a combination of Chern numbers, not a restatement of the input. The proof of the quoted slope estimate is not completed in Appendix A (Claim A.6: 'The estimates of Term (I) and Term (III) are the same as in [DGP24, The term (I) and (III)]. We therefore omit the proof.'), but an omitted proof is a verifiability gap, not a circular loop. The self-citations [IMM25, IJZ25] are co-authored by Iwai but are published, parameter-free external results and are not used to assert the paper's own conclusion; the central claim does not reduce to them by construction. Theorem 1.2 has the same architecture: it imports [HL26, Theorems 7.1/7.2] and uses [DJ25, Theorem 1.2] for δ'_v; the paper's own equivariant Bogomolov–Gieseker inequality (Corollary 5.2) is stated with proof omitted, but it is presented as a consequence of Theorem 4.3 by a limiting argument, not as a disguised form of the Miyaoka–Yau inequality. No fitted parameter is renamed as a prediction and no uniqueness theorem is imported from the authors.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No parameters are fitted to data: the δ-invariant is an input invariant; the soliton parameter a ≈ −0.5276195 in Example 6.3 is determined by the soliton equation, not chosen to make the inequality hold. No new postulates in the physicists' sense are introduced: the new objects (v-slopes, weighted equivariant intersection numbers, the canonical extension with class √(2πν)c₁) are explicitly defined constructions, not entities invoked to explain the result. The heaviest load is carried by the two external slope estimates (XZ26 for klt, HL26 for solitons) and by the published klt Langer inequality (IMM25).

assumptions (7)
  • domain assumption Assumption 2.1: quotient (X×U)/G exists as a scheme for the G-actions considered; for tori this follows from condition (3) of [EG98, Prop. 23].
    Standing technical assumption for the equivariant-Chow machinery; standard for torus actions but stated, not proved.
  • ad hoc to paper klt Tian slope estimate for the canonical extension sheaf V: µ_min(V) ≥ δ′(X)/(n+1)·c₁(X)^n (Theorem 7.1 = [XZ26, Lemma 5.9]).
    Core input of Theorem 1.1, imported from a contemporaneous preprint; Appendix A's alternative proof omits the key asymptotic estimates (Claim A.6, Terms I and III).
  • ad hoc to paper Soliton slope estimates for the canonical extension sheaf under Ric_v ω ≥ γω (Theorem 6.2 = [HL26, Theorems 7.1–7.2]).
    Core input of Theorem 1.2; quoted from Hallam–Lahdili with a reduction proof.
  • domain assumption δ′(X) is the greatest Ricci lower bound ([CRZ19, BBJ21]); δ′_v(X) is the greatest weighted Ricci lower bound ([DJ25, Theorem 1.2], eq. (1.3)).
    Converts the algebraic δ-invariant into the curvature condition needed by the analytic slope estimates.
  • domain assumption Canonical normalization of v: (c₁^T(X)·e^{ĉ₁(X)})^T(ξ) = n(e^{ĉ₁(X)})^T(ξ), eq. (1.2).
    Explicit hypothesis of Theorem 1.2; automatic for Kähler–Ricci soliton moment maps.
  • domain assumption Orbifold second Chern class Ĉ₂(X) is well-defined for klt varieties ([Kaw92, GKPT19, GK20]).
    Needed for the statement of Theorem 1.1 on singular varieties.
  • domain assumption Langer's inequality for klt varieties ([IMM25, Proposition 3.4]).
    Published with proof; converts the slope gap into the Chern-number bound; co-authored by one of the present authors but independently established.

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Pith. "Pith review of The Miyaoka-Yau inequality and the delta invariant for Fano varieties." pith.science (2026). https://pith.science/paper/QKNCI4NL

@misc{pith2026260725181,
  author       = {Pith},
  title        = {Pith review of: The Miyaoka-Yau inequality and the delta invariant for Fano varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QKNCI4NL}},
  note         = {Machine review of arXiv:2607.25181}
}
abstract

We establish the following Miyaoka-Yau inequality for any $n$-dimensional klt Fano variety $X$, possibly K-unstable, in terms of its delta invariant: $$ \left(2(n+1)\widehat{c}_2(X)-n c_1(X)^2\right)\cdot c_1(X)^{n-2} \ge -n \left(1-\min\{1,\delta(X)\}\right)^2 \cdot c_1(X)^n. $$ Furthermore, inspired by recent work of Inoue and Hallam-Lahdili, we formulate and prove an equivariant version of this inequality in the soliton setting, in which the Chern classes and the delta invariant are replaced by the equivariant Chern classes and the weighted delta invariant, respectively. As a consequence, we obtain the equivariant Miyaoka-Yau inequality for every smooth Fano manifold admitting a K\"ahler-Ricci soliton.

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