Paper · Feb 15, 2018

Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle

Ryosuke Nomura

math.DGarXiv:1802.05425
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Abstract

In this paper, we prove the Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle. The key point of the proof is the estimate for the -norm of the scalar curvature along the Kähler-Ricci flow.

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Submitted
Feb 15, 2018
Subjects
math.DG, math.AG, math.CV

Comments: 5pages

Ricci flowKähler–Ricci flowKähler geometryScalar curvature

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Blow-up rate of the scalar curvature along the conical Kähler-Ricci flow with finite time singularities (Ryosuke Nomura, 2016)3Nomura 2016 Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach (Ryosuke Nomura, 2016)2Nomura 2016 Cusp Kähler-Ricci flow on compact Kähler manifold (Jiawei Liu and Xi Zhang, 2017)6Liu 2017 Immortal solutions of the Kähler-Ricci flow (Valentino Tosatti, 2024)8Tosatti 2024 Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold (Lei Zhang and Zhenlei Zhang, 2026)4Zhang 2026 The Miyaoka-Yau inequality and the delta invariant for Fano varieties (Tomoyuki Hisamoto and Masataka Iwai, 2026)7Hisamoto 2026 On Miyaoka-Yau Inequalities and Weil-Petersson Metrics (Alexander Bednarek, 2026)1Bednarek 2026 The Miyaoka-Yau inequality and the delta invariant for Fano varieties II -soliton case- (Tomoyuki Hisamoto and Masataka Iwai, 2026)5Hisamoto 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On Miyaoka-Yau Inequalities and Weil-Petersson Metrics

    Alexander Bednarek

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “Miyaoka Yau”, “Yau inequality”, “compact Kähler”

  2. Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach

    Ryosuke Nomura

    Topics Kähler–Ricci flow, Kähler geometry · Same author Ryosuke Nomura · Shared terms “canonical bundle”, “compact Kähler”, “Yau”

  3. Blow-up rate of the scalar curvature along the conical Kähler-Ricci flow with finite time singularities

    Ryosuke Nomura

    Topics Kähler–Ricci flow, Scalar curvature, Kähler geometry · Same author Ryosuke Nomura · Shared terms “curvature along”

  4. Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold

    Lei Zhang and Zhenlei Zhang

    Topics Kähler–Ricci flow, Scalar curvature, Kähler geometry · Shared terms “canonical bundle”, “compact Kähler”

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

    Tomoyuki Hisamoto and Masataka Iwai

    Topics Kähler geometry · Shared terms “Miyaoka Yau”, “Yau inequality”

  6. Cusp Kähler-Ricci flow on compact Kähler manifold

    Jiawei Liu and Xi Zhang

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “compact Kähler”, “canonical bundle”

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

    Tomoyuki Hisamoto and Masataka Iwai

    Topics Kähler geometry · Shared terms “Miyaoka Yau”, “Yau inequality”

  8. Immortal solutions of the Kähler-Ricci flow

    Valentino Tosatti

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “compact Kähler”

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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  3. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.