Paper · Oct 11, 2016

The Kähler-Ricci flow on Fano bundles

Xin Fu, Shijin Zhang

math.DGarXiv:1610.03366v2
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Abstract

We study the behavior of the Kähler-Ricci flow on some Fano bundle which is a trivial bundle on one Zariski open set. We show that if the fiber is blown up at one point or some weighted projective space blown up at the orbifold point and the initial metric is in a suitable kähler class, then the fibers collapse in finite time and the metrics converge sub-sequentially in Gromov-Hausdorff sense to a metric on the base.

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Submitted
Oct 11, 2016
Last revised
Dec 7, 2016 (v2)
Subjects
math.DG

Comments: Rewrite some part of the introduction, add more references, and correct the computation in the proof of Lemma 2.2. Welcome any Comments. arXiv admin note: text overlap with arXiv:1003.0718 by other authors

Ricci flowKähler–Ricci flowSingularitiesKähler geometryLimit spaces & RCD

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 Kahler-Ricci flow on blowups along submanifolds (Bin Guo, 2018)3Guo 2018 The Kähler-Ricci flow, holomorphic vector fields and Fano bundles (Xi Sisi Shen, 2020)1Shen 2020 Horosymmetric limits of Kähler-Ricci flow on Fano G-manifolds (Gang Tian and Xiaohua Zhu, 2022)5Tian 2022 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)4Hallgren 2023 Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow (Junsheng Zhang, 2025)8Zhang 2025 Kähler-Ricci Flow: from divisors to cusps (Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu, 2026)6Di Nezza 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles (Wangjian Jian and Jian Song, 2026)2Jian 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II (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. The Kähler-Ricci flow, holomorphic vector fields and Fano bundles

    Xi Sisi Shen

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

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

    Wangjian Jian and Jian Song

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

  3. Kahler-Ricci flow on blowups along submanifolds

    Bin Guo

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Shared terms “sequentially”, “Gromov Hausdorff”

  4. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al.

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Shared terms “Gromov Hausdorff”

  5. Horosymmetric limits of Kähler-Ricci flow on Fano -manifolds

    Gang Tian and Xiaohua Zhu

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Shared terms “Fano”, “initial metric”, “Gromov Hausdorff”

  6. Kähler-Ricci Flow: from divisors to cusps

    Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “Zariski open”, “open set”

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

    Wangjian Jian and Jian Song

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

  8. Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow

    Junsheng Zhang

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

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.