Paper · Feb 22, 2018

The moduli space of Fano manifolds with Kähler-Ricci solitons

Eiji Inoue

math.DGarXiv:1802.08128v4
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Abstract

We construct a canonical Hausdorff complex analytic moduli space of Fano manifolds with Kähler-Ricci solitons. This naturally enlarges the moduli space of Fano manifolds with Kähler-Einstein metrics, which was constructed by Odaka and Li-Wang-Xu. We discover a moment map picture for Kähler-Ricci solitons, and give complex analytic charts on the topological space consisting of Kähler-Ricci solitons, by studying differential geometric aspects of this infinite dimensional moment map. Some stacky words and arguments on Gromov-Hausdorff convergence help to glue them together in the holomorphic manner.

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Submitted
Feb 22, 2018
Last revised
Apr 14, 2020 (v4)
Subjects
math.DG, math.AG

Comments: final version, 54 pages

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryLimit spaces & RCDKähler–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 Compactness of Kahler-Ricci solitons on Fano manifolds (Bin Guo, Duong H. Phong, Jian Song et al., 2018)6Guo 2018 Existence of Kahler-Ricci solitons on smoothable Q-Fano varities (Yan Li, 2019)3Li 2019 Examples of K-unstable Fano manifolds (Thibaut Delcroix, 2019)5Delcroix 2019 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)1Guo 2020 Equivariant calculus on μ-character and μK-stability of polarized schemes (Eiji Inoue, 2020)4Inoue 2020 Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold (Feng Wang and Xiaohua Zhu, 2020)7Wang 2020 Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type (Yasuhiro Nakagawa and Satoshi Nakamura, 2022)8Nakagawa 2022 Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein (Yasufumi Nitta, 2026)2Nitta 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 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”

  2. 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”

  3. Existence of Kahler-Ricci solitons on smoothable Q-Fano varities

    Yan Li

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Limit spaces & RCD · Shared terms “Fano”, “Gromov Hausdorff”

  4. Equivariant calculus on -character and K-stability of polarized schemes

    Eiji Inoue

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Same author Eiji Inoue · Shared terms “geometric aspects”, “infinite”

  5. Examples of K-unstable Fano manifolds

    Thibaut Delcroix

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

  6. 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, Limit spaces & RCD · Shared terms “Fano manifolds”, “Gromov Hausdorff”

  7. 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”

  8. Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type

    Yasuhiro Nakagawa and Satoshi Nakamura

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

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. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.