Paper · Jul 28, 2026

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

Tomoyuki Hisamoto, Masataka Iwai

math.AGarXiv:2607.25181v2
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Abstract

We establish the following Miyaoka-Yau inequality for any -dimensional klt Fano variety , possibly K-unstable, in terms of its delta invariant: 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 Kähler-Ricci soliton.

Comments: v2: 66 pages; Minor revision. Theorem 1.1 and Theorem B.3 have been improved. v1: 64 pages; comments are welcome

Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniquenessKähler–Einstein & K-stabilityAnalytic minimal model program

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  1. The Miyaoka-Yau inequality and the delta invariant for Fano varieties II -soliton case-

    Tomoyuki Hisamoto and Masataka Iwai PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Same authors Tomoyuki Hisamoto, Masataka Iwai · Shared terms “Miyaoka Yau”, “Yau inequality”, “delta invariant”

  2. On Miyaoka-Yau Inequalities and Weil-Petersson Metrics

    Alexander Bednarek PDF

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler geometry · Shared terms “Miyaoka Yau”, “Yau inequality”, “Fano”

  3. K-stability of Gorenstein Fano group compactifications with rank two

    Jae-Hyouk Lee, Kyeong-Dong Park and Sungmin Yoo PDF

    Topics Kähler–Einstein & K-stability, Kähler geometry, Stability & uniqueness · Shared terms “delta invariant”, “Fano varieties”, “equivariant”

  4. Kähler-Ricci solitons on Fano threefolds with non-trivial moduli

    Minghao Miao and Linsheng Wang PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “Fano varieties”, “admitting”

  5. Shrinking Kähler-Ricci solitons on toric fano fibrations

    Tristan C. Collins, Genggeng Huang, Freid Tong et al. PDF

    Topics Analytic minimal model program, Kähler–Ricci solitons, Kähler geometry · Shared terms “Fano”

  6. From Kähler Ricci solitons to Calabi-Yau Kähler cones

    Vestislav Apostolov, Abdellah Lahdili and Eveline Legendre PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “smooth Fano”, “Yau”, “Fano manifold”

  7. Optimal Degenerations of K-unstable Fano threefolds

    Minghao Miao and Linsheng Wang PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “Fano varieties”, “unstable”, “admitting”

  8. Shrinking gradient Kähler-Ricci solitons are simply-connected

    Carlos Esparza PDF

    Topics Analytic minimal model program, Kähler–Ricci solitons, Kähler geometry · Shared terms “Fano”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.