Paper · Nov 17, 2019

Geometry of twisted Kähler-Einstein metrics and collapsing

Mark Gross, Valentino Tosatti, Yuguang Zhang

math.DGarXiv:1911.07315v2
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Abstract

We prove that the twisted Kahler-Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi-Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when studying the collapsing of Ricci-flat Kahler metrics on Calabi-Yau manifolds, and of the Kahler-Ricci flow on compact Kahler manifolds with semiample canonical bundle and intermediate Kodaira dimension. Our results allow us to understand their collapsed Gromov-Hausdorff limits when the base is smooth and the discriminant has simple normal crossings.

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Submitted
Nov 17, 2019
Last revised
Nov 20, 2020 (v2)
Subjects
math.DG, math.AG, math.MG
Journal
Comm. Math. Phys. 380 (2020), no.3, 1401-1438
DOI
10.1007/s00220-020-03911-0

Comments: 39 pages; final version to appear in Comm Math Phys

Ricci flowKähler–Ricci flowEinstein metricsRicci-flat & Calabi–YauSingularities(Non)collapsingKähler geometryLimit spaces & RCDKähler–Einstein & K-stabilityAnalytic minimal model program

Its neighbourhood

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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)8Zhang 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)5Song 2019 Twisted Kähler-Einstein Metrics and Collapsing (Kyle Broder, 2020)2Broder 2020 On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows (Yang Li and Valentino Tosatti, 2021)1Li 2021 Collapsing immortal Kähler-Ricci flows (Hans-Joachim Hein, Man-Chun Lee and Valentino Tosatti, 2024)4Hein 2024 Fano Fibrations and Twisted Kähler-Einstein Metrics I (Alexander Bednarek, 2025)6Bednarek 2025 Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow (Alexander Bednarek, 2025)7Bednarek 2025 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)3Lee 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows

    Yang Li and Valentino Tosatti

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

  2. Twisted Kähler-Einstein Metrics and Collapsing

    Kyle Broder

    Topics Ricci-flat & Calabi–Yau, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “twisted Kähler”, “Hausdorff limit”, “Gromov Hausdorff”

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

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Same author Valentino Tosatti · Shared terms “twisted Kähler”, “Hausdorff limit”, “semiample canonical”

  4. Collapsing immortal Kähler-Ricci flows

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

    Topics Analytic minimal model program, Ricci-flat & Calabi–Yau, Kähler–Ricci flow · Same author Valentino Tosatti · Shared terms “intermediate Kodaira”, “semiample canonical”, “Kodaira dimension”

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

    Jian Song, Gang Tian and Zhenlei Zhang

    Topics Analytic minimal model program, Ricci-flat & Calabi–Yau, Kähler–Einstein & K-stability · Shared terms “twisted Kähler”, “Kodaira dimension”, “canonical bundle”

  6. Fano Fibrations and Twisted Kähler-Einstein Metrics I

    Alexander Bednarek

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “twisted Kähler”, “compact Kähler”, “studying”

  7. Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow

    Alexander Bednarek

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “twisted Kähler”, “compact Kähler”, “studying”

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

    Yashan Zhang

    Topics Analytic minimal model program, Ricci-flat & Calabi–Yau, Kähler–Einstein & K-stability · Shared terms “Gromov Hausdorff”

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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)

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.