Paper · Mar 31, 2020

The Kähler-Ricci flow, holomorphic vector fields and Fano bundles

Xi Sisi Shen

math.DGarXiv:2004.00042
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Abstract

We study the behavior of the Kähler-Ricci flow on compact manifolds developing finite-time singularities, in particular, when the flow contracts exceptional divisors or collapses the Fano fibers of a holomorphic fiber bundle. We present a technique using holomorphic vector fields to prove estimates related to the work of Song-Weinkove and Fu-Zhang.

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Submitted
Mar 31, 2020
Subjects
math.DG

Comments: 17 pages

Ricci flowKähler–Ricci flowSingularitiesKähler geometryAnalytic minimal model program

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 The Kähler-Ricci flow on Fano bundles (Xin Fu and Shijin Zhang, 2016)2Fu 2016 Infinite-time singularity type of the Kähler-Ricci flow (Yashan Zhang, 2017)6Zhang 2017 Infinite-time singularity type of the Kähler-Ricci flow II (Yashan Zhang, 2018)5Zhang 2018 On almost nonpositive k-Ricci curvature (Kai Tang, 2021)8Tang 2021 Finite Time Type I Singularities of the Kähler Ricci Flow (Tongxin Xu and Zhenlei Zhang, 2026)3Xu 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles (Wangjian Jian and Jian Song, 2026)1Jian 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II (Wangjian Jian and Jian Song, 2026)4Jian 2026 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)7Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles

    Wangjian Jian and Jian Song

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “Fano bundle”, “time singularity”, “finite time”

  2. The Kähler-Ricci flow on Fano bundles

    Xin Fu and Shijin Zhang

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “Fano bundle”, “collapse”, “finite time”

  3. 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 “exceptional divisor”, “finite time”

  4. Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II

    Wangjian Jian and Jian Song

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “Fano bundle”, “time singularity”, “finite time”

  5. Infinite-time singularity type of the Kähler-Ricci flow II

    Yashan Zhang

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

  6. Infinite-time singularity type of the Kähler-Ricci flow

    Yashan Zhang

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

  7. 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 “time singularity”, “finite time”

  8. On almost nonpositive -Ricci curvature

    Kai Tang

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

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.