Paper · Jan 26, 2021

Balanced metrics for Kähler-Ricci solitons and quantized Futaki invariants

Louis Ioos

math.DGarXiv:2101.10974v2
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Abstract

We show that a Kähler-Ricci soliton on a Fano manifold can always be smoothly approximated by a sequence of relative anticanonically balanced metrics, also called quantized Kähler-Ricci solitons. The proof uses a semiclassical estimate on the spectral gap of an equivariant Berezin transform to extend a strategy due to Donaldson, and can be seen as the quantization of a method due to Tian and Zhu, using quantized Futaki invariants as obstructions for quantized Kähler-Ricci solitons. As corollaries, we recover the uniqueness of Kähler-Ricci solitons up to automorphisms, and show how our result also applies to Kähler-Einstein Fano manifolds with general automorphism group.

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Submitted
Jan 26, 2021
Last revised
Apr 26, 2022 (v2)
Subjects
math.DG, math.CV
Journal
J. Funct. Anal. 282 (2022) no. 8, 109400

Comments: 53 pages, published version

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryStability & uniquenessEigenvalues & spectrumKähler–Einstein & K-stability

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 K-Stability of Fano spherical varieties (Thibaut Delcroix, 2016)6Delcroix 2016 The Kähler-Ricci flow and optimal degenerations (Ruadhaí Dervan and Gábor Székelyhidi, 2016)7Dervan 2016 Compactness of Kahler-Ricci solitons on Fano manifolds (Bin Guo, Duong H. Phong, Jian Song et al., 2018)8Guo 2018 The Kähler-Ricci flow and quantitative bounds for Donaldson-Futaki invariants of optimal degenerations (Ryosuke Takahashi, 2019)2Takahashi 2019 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)3Guo 2020 Basis divisors and balanced metrics (Yanir A. Rubinstein, Gang Tian and Kewei Zhang, 2020)1Rubinstein 2020 Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold (Feng Wang and Xiaohua Zhu, 2020)4Wang 2020 Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein (Yasufumi Nitta, 2026)5Nitta 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Basis divisors and balanced metrics

    Yanir A. Rubinstein, Gang Tian and Kewei Zhang

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Shared terms “balanced metrics”, “quantized”, “automorphisms”

  2. The Kähler-Ricci flow and quantitative bounds for Donaldson-Futaki invariants of optimal degenerations

    Ryosuke Takahashi

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Shared terms “Futaki invariant”, “Fano manifolds”, “Donaldson”

  3. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Fano manifolds”

  4. Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Fano manifolds”, “Zhu”, “Tian”

  5. Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein

    Yasufumi Nitta

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Fano manifolds”

  6. K-Stability of Fano spherical varieties

    Thibaut Delcroix

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Fano manifolds”, “equivariant”, “Donaldson”

  7. The Kähler-Ricci flow and optimal degenerations

    Ruadhaí Dervan and Gábor Székelyhidi

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Fano manifolds”, “automorphisms”, “Zhu”

  8. Compactness of Kahler-Ricci solitons on Fano manifolds

    Bin Guo, Duong H. Phong, Jian Song et al.

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Shared terms “Futaki invariant”, “Fano manifolds”, “sequence”

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) 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) 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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.