Paper · Oct 27, 2017

An approach to Griffiths conjecture

Philipp Naumann

math.AGarXiv:1710.10034
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Abstract

The Griffiths conjecture asserts that every ample vector bundle over a compact complex manifold admits a hermitian metric with positive curvature in the sense of Griffiths. In this article we give a sufficient condition for a positive hermitian metric on to induce a Griffiths positive -metric on the vector bundle . This result suggests to study the relative Kähler-Ricci flow on for the fibration . We define a flow and give arguments for the convergence.

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Submitted
Oct 27, 2017
Subjects
math.AG, math.CV, math.DG

Comments: 12 pages

Ricci flowKähler–Ricci flowKähler geometry

Its neighbourhood

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A note on compact -solutions of Kähler-Ricci flow

    Yuxing Deng and Xiaohua Zhu

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

  2. Positivity preserving along a flow over projective bundle

    Xueyuan Wan

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

  3. Griffiths positivity for Bismut curvature and its behaviour along Hermitian Curvature Flows

    Giuseppe Barbaro

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

  4. Kähler-Ricci flow of cusp singularities on quasi projective varieties

    Albert Chau, Ka-Fai Li and Liangming Shen

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

  5. An almost complex Chern-Ricci flow

    Tao Zheng

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “Hermitian metric”, “complex manifold”

  6. Kähler manifolds and cross quadratic bisectional curvature

    Lei Ni and Fangyang Zheng

    Topics Kähler–Ricci flow, Kähler geometry

  7. Geometric smoothing by the Kähler-Ricci Flow

    Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu

    Topics Kähler–Ricci flow, Kähler geometry

  8. Kähler-Ricci flow and conformal submersion

    Hoan Nguyen

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

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.