Paper · Jul 2, 2021

On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows

Yang Li, Valentino Tosatti

math.DGarXiv:2107.00836v3
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Abstract

We study the collapsing of Calabi-Yau metrics and of Kahler-Ricci flows on fiber spaces where the base is smooth. We identify the collapsed Gromov-Hausdorff limit of the Kahler-Ricci flow when the divisorial part of the discriminant locus has simple normal crossings. In either setting, we also obtain an explicit bound for the real codimension 2 Hausdorff measure of the Cheeger-Colding singular set, and identify a sufficient condition from birational geometry to understand the metric behavior of the limiting metric on the base.

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Submitted
Jul 2, 2021
Last revised
Aug 7, 2024 (v3)
Subjects
math.DG, math.AG, math.MG
Journal
J. Reine Angew. Math. 800 (2023), 155-192
DOI
10.1515/crelle-2023-0025

Comments: 37 pages; final version to appear in J. Reine Angew. Math; typos fixed

Ricci flowKähler–Ricci flowRicci-flat & Calabi–Yau(Non)collapsingKähler 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 Collapsing limits of the Kähler-Ricci flow and the continuity method (Yashan Zhang, 2017)5Zhang 2017 Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow (Jian Song, Gang Tian and Zhenlei Zhang, 2019)7Song 2019 A "boundedness implies convergence" principle and its applications to collapsing estimates in Kähler geometry (Wangjian Jian and Yalong Shi, 2019)8Jian 2019 Geometry of twisted Kähler-Einstein metrics and collapsing (Mark Gross, Valentino Tosatti and Yuguang Zhang, 2019)1Gross 2019 Twisted Kähler-Einstein Metrics and Collapsing (Kyle Broder, 2020)3Broder 2020 Collapsing immortal Kähler-Ricci flows (Hans-Joachim Hein, Man-Chun Lee and Valentino Tosatti, 2024)2Hein 2024 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)6Lee 2026 Parabolic Monge-Ampère equations on a family of Calabi-Yau manifolds (Jiyuan Han, 2026)4Han 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Geometry of twisted Kähler-Einstein metrics and collapsing

    Mark Gross, Valentino Tosatti and Yuguang Zhang

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, (Non)collapsing · Same author Valentino Tosatti · Shared terms “fiber spaces”, “discriminant”, “collapsed Gromov”

  2. Collapsing immortal Kähler-Ricci flows

    Hans-Joachim Hein, Man-Chun Lee and Valentino Tosatti

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, (Non)collapsing · Same author Valentino Tosatti · Shared terms “Yau metrics”

  3. Twisted Kähler-Einstein Metrics and Collapsing

    Kyle Broder

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, (Non)collapsing · Shared terms “Hausdorff limit”, “Gromov Hausdorff”

  4. Parabolic Monge-Ampère equations on a family of Calabi-Yau manifolds

    Jiyuan Han

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Limit spaces & RCD · Shared terms “Hausdorff”, “Gromov”

  5. Collapsing limits of the Kähler-Ricci flow and the continuity method

    Yashan Zhang

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, (Non)collapsing · Shared terms “Gromov Hausdorff”

  6. Gromov-Hausdorff limits of immortal Kähler-Ricci flows

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same author Valentino Tosatti · Shared terms “Hausdorff limit”, “Gromov Hausdorff”

  7. Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow

    Jian Song, Gang Tian and Zhenlei Zhang

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, (Non)collapsing · Shared terms “Gromov Hausdorff”, “Hausdorff limit”

  8. A "boundedness implies convergence" principle and its applications to collapsing estimates in Kähler geometry

    Wangjian Jian and Yalong Shi

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, (Non)collapsing · Shared terms “Yau metrics”

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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.