Paper · Oct 6, 2016

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

Ryosuke Nomura

math.DGarXiv:1610.01976v2
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Abstract

Recently, Wu-Yau and Tosatti-Yang established the connection between the negativity of holomorphic sectional curvatures and the positivity of canonical bundles for compact Kähler manifolds. In this short note, we give anothe proof of their theorems by using the Kähler-Ricci flow.

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Submitted
Oct 6, 2016
Last revised
Feb 15, 2018 (v2)
Subjects
math.DG, math.CV
Journal
International Mathematics Research Notices, rnx075, 2017
DOI
10.1093/imrn/rnx075

Comments: 4pages, minor changes

Ricci flowKähler–Ricci flowKähler geometry

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)5Nomura 2016 Hirzebruch manifolds and positive holomorphic sectional curvature (Bo Yang and Fangyang Zheng, 2016)4Yang 2016 Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle (Ryosuke Nomura, 2018)2Nomura 2018 The Kähler-Ricci flow on manifolds with negative holomorphic curvature (Freid Tong, 2018)1Tong 2018 Longtime existence of Kähler Ricci flow and holomorphic sectional curvature (Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al., 2018)3Huang 2018 Complete Kähler-Einstein metric on Stein manifolds with negative curvature (Man-Chun Lee, 2019)8Lee 2019 On almost nonpositive k-Ricci curvature (Kai Tang, 2021)6Tang 2021 Immortal solutions of the Kähler-Ricci flow (Valentino Tosatti, 2024)7Tosatti 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Kähler-Ricci flow on manifolds with negative holomorphic curvature

    Freid Tong

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “negative holomorphic”, “holomorphic sectional”, “sectional curvature”

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

    Ryosuke Nomura

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

  3. Longtime existence of Kähler Ricci flow and holomorphic sectional curvature

    Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al.

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “holomorphic sectional”, “sectional curvature”, “Yau”

  4. Hirzebruch manifolds and positive holomorphic sectional curvature

    Bo Yang and Fangyang Zheng

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

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

    Ryosuke Nomura

    Topics Kähler–Ricci flow, Kähler geometry · Same author Ryosuke Nomura

  6. On almost nonpositive -Ricci curvature

    Kai Tang

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

  7. Immortal solutions of the Kähler-Ricci flow

    Valentino Tosatti

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

  8. Complete Kähler-Einstein metric on Stein manifolds with negative curvature

    Man-Chun Lee

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “holomorphic sectional”, “sectional curvature”

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)

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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.