Paper · Oct 28, 2026

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

Tomoyuki Hisamoto, Masataka Iwai

math.AGarXiv:2610.38137
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Abstract

Inspired by recent work of Inoue and Hallam-Lahdili, we prove an equivariant Miyaoka-Yau inequality with a correction term for smooth Fano manifolds admitting Kähler-Ricci solitons. It recovers the usual Miyaoka-Yau inequality in the Kähler-Einstein case. Our proof uses weighted equivariant Bogomolov-Gieseker and Langer inequalities for torsion-free sheaves, together with a weighted slope estimate for the canonical extension sheaf. We also construct toric examples showing that the existence of a soliton implies neither weighted semistability of the canonical extension sheaf nor nonnegativity of the equivariant Miyaoka-Yau number.

Comments: v1: 54 pages; An earlier version of this paper was included as the second part of arXiv:2607.25181. The original manuscript has since been divided into two independent papers. Comments are welcome

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryKähler–Einstein & K-stability

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    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”

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    Topics Kähler–Einstein & K-stability, Kähler geometry · Shared terms “Miyaoka Yau”, “Yau inequality”, “Fano manifold”

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

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

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

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

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  8. An Aubin continuity path for shrinking gradient Kähler-Ricci solitons

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

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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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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)

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.