Paper · Aug 23, 2021

On almost nonpositive -Ricci curvature

Kai Tang

math.DGarXiv:2108.10237
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Abstract

Motivated by the recent work of Chu-Lee-Tam on the nefness of canonical line bundle for compact Kähler manifolds with nonpositive -Ricci curvature, we consider a natural notion of {\em almost nonpositive -Ricci curvature}, which is weaker than the existence of a Kähler metric with nonpositive -Ricci curvature. When , this is just the {\em almost nonpositive holomorphic sectional curvature} introduced by Zhang. We firstly give a lower bound for the existence time of the twisted Kähler-Ricci flow when there exists a Kähler metric with -Ricci curvature bounded from above by a positive constant. As an application, we prove that a compact Kähler manifold of almost nonpositive -Ricci curvature must have nef canonical line bundle.

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Submitted
Aug 23, 2021
Subjects
math.DG

Comments: 10 pages

Ricci flowKähler–Ricci flowKähler geometryAnalytic minimal model program

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  1. Fano Fibrations and Twisted Kähler-Einstein Metrics I

    Alexander Bednarek

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “twisted Kähler”, “line bundle”, “canonical line”

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  3. Scalar Curvature, Volumes and the Bergman Kernel

    Bo-Yong Chen, Yuanpu Xiong and Liyou Zhang

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “canonical line”, “line bundle”, “nef canonical”

  4. Curvature Estimates for the Continuity Method

    Hosea Wondo

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “canonical line”, “line bundle”, “compact Kähler”

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

    Lei Zhang and Zhenlei Zhang

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “compact Kähler”

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

    Alexander Bednarek

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “canonical line”, “line bundle”, “compact Kähler”

  7. Finite Time Type I Singularities of the Kähler Ricci Flow

    Tongxin Xu and Zhenlei Zhang

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “compact Kähler”

  8. Type I estimates for the Kahler-Ricci flow I

    Wangjian Jian and Jian Song

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “compact Kähler”

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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 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. 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) Analytic minimal model program On a projective variety the Kähler–Ricci flow runs until its Kähler class stops being Kähler, a time read off from cohomology. Start with: Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type (Tsuji, 1988)

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.