Paper · Dec 11, 2023

Kähler-Ricci Tangent Flows are Infinitesimally Algebraic

Max Hallgren

math.DGarXiv:2312.06577v2
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Abstract

We show that any tangent cone of a singular shrinking Kähler-Ricci soliton is a normal affine algebraic variety. Moreover, the regular set of such a tangent cone in the metric sense coincides with the regular set in the algebraic sense. Along the way, we give a parabolic proof of Hörmander's estimate, which can be used to solve the -equation on any singular shrinking Kähler-Ricci soliton.

Comments: Minor corrections and simplifications

Ricci solitonsKähler–Ricci solitonsSingularitiesKähler 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 2021, 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 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)4Hallgren 2023 On Kähler-Einstein Currents (Yifan Chen, Shih-Kai Chiu, Max Hallgren et al., 2025)5Chen 2025 Toric geometry of generalized Kähler-Ricci solitons (Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al., 2025)6Apostolov 2025 A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity (Richard H. Bamler, Eric Chen and Ronan J. Conlon, 2026)1Bamler 2026 Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2026)3Conlon 2026 Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program (Longteng Chen, Max Hallgren and Lucas Lavoyer, 2026)2Chen 2026 Shrinking Kähler-Ricci solitons on toric fano fibrations (Tristan C. Collins, Genggeng Huang, Freid Tong et al., 2026)8Collins 2026 Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton (Ronan J. Conlon and Alix Deruelle, 2026)7Conlon 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity

    Richard H. Bamler, Eric Chen and Ronan J. Conlon PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “tangent cone”

  2. Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program

    Longteng Chen, Max Hallgren and Lucas Lavoyer PDF

    Topics Kähler geometry, Singularities, Ricci solitons · Same author Max Hallgren · Shared terms “ricci tangent”, “tangent flow”

  3. Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

    Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “shrinking Kähler”

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

    Max Hallgren, Wangjian Jian, Jian Song et al. PDF

    Topics Kähler geometry, Singularities · Same author Max Hallgren · Shared terms “tangent cone”

  5. On Kähler-Einstein Currents

    Yifan Chen, Shih-Kai Chiu, Max Hallgren et al. PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author Max Hallgren

  6. Toric geometry of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al. PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons

  7. Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton

    Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons

  8. Shrinking Kähler-Ricci solitons on toric fano fibrations

    Tristan C. Collins, Genggeng Huang, Freid Tong et al. PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “shrinking Kähler”

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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, 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. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

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  2. 6C.8
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    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 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.