Paper · Jan 30, 2018

Positivity preserving along a flow over projective bundle

Xueyuan Wan

math.DGarXiv:1801.09886v2
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Abstract

In this paper, we introduce a flow over the projective bundle , which is a natural generalization of both Hermitian-Yang-Mills flow and Kähler-Ricci flow. We prove that the semipositivity of curvature of the hyperplane line bundle is preserved along this flow under the null eigenvector assumption. As applications, we prove that the semipositivity is preserved along the this flow if the base manifold is a curve, which implies that the Griffiths semipositivity is preserved along the Hermitian-Yang-Mills flow over a curve. And we also reprove that the nonnegativity of holomorphic bisectional curvature is preserved under Kähler-Ricci flow.

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Submitted
Jan 30, 2018
Last revised
Apr 29, 2021 (v2)
Subjects
math.DG, math.AP

Comments: Final version, to appear in Communications in Analysis and Geometry

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 Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach (Ryosuke Nomura, 2016)2Nomura 2016 The Kähler-Ricci flow on Fano bundles (Xin Fu and Shijin Zhang, 2016)7Fu 2016 Kähler-Ricci flow of cusp singularities on quasi projective varieties (Albert Chau, Ka-Fai Li and Liangming Shen, 2017)6Chau 2017 An approach to Griffiths conjecture (Philipp Naumann, 2017)1Naumann 2017 A note on compact κ-solutions of Kähler-Ricci flow (Yuxing Deng and Xiaohua Zhu, 2018)8Deng 2018 Kähler manifolds and cross quadratic bisectional curvature (Lei Ni and Fangyang Zheng, 2019)4Ni 2019 Kähler-Ricci Flow preserves negative anti-bisectional curvature (Gabriel Khan and Fangyang Zheng, 2020)5Khan 2020 Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation (Randall R. Van Why, 2026)3Van Why 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. An approach to Griffiths conjecture

    Philipp Naumann

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

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

    Ryosuke Nomura

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “positivity”, “holomorphic”, “Yang”

  3. Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation

    Randall R. Van Why

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

  4. Kähler manifolds and cross quadratic bisectional curvature

    Lei Ni and Fangyang Zheng

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “bisectional curvature”, “positivity”

  5. Kähler-Ricci Flow preserves negative anti-bisectional curvature

    Gabriel Khan and Fangyang Zheng

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “bisectional curvature”, “preserved”

  6. 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 “projective”, “bisectional curvature”, “Griffiths”

  7. The Kähler-Ricci flow on Fano bundles

    Xin Fu and Shijin Zhang

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

  8. A note on compact -solutions of Kähler-Ricci flow

    Yuxing Deng and Xiaohua Zhu

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

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.