Paper · Sep 5, 2019

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

Ryosuke Takahashi

math.DGarXiv:1909.02452v3
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Abstract

We establish a lower bound for the Donaldson-Futaki invariant of optimal degenerations produced by the Kähler-Ricci flow in terms of the greatest Ricci lower bound on arbitrary Fano manifolds. As an application, we can generalize the finiteness of the Futaki invariants on Kähler-Ricci solitons obtained by Guo-Phong-Song-Sturm to the space of all Fano manifolds. Also, we discuss the relation to Hisamoto's inequality for the infimum of the -functional.

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Submitted
Sep 5, 2019
Last revised
Jan 17, 2020 (v3)
Subjects
math.DG

Comments: 11 pages, final version, to appear in Proc. Amer. Math. Soc

Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsKähler geometryKähler–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 The Kähler-Ricci flow and optimal degenerations (Ruadhaí Dervan and Gábor Székelyhidi, 2016)2Dervan 2016 Compactness of Kahler-Ricci solitons on Fano manifolds (Bin Guo, Duong H. Phong, Jian Song et al., 2018)1Guo 2018 Geometric flow, Multiplier ideal sheaves and Optimal destabilizer for a Fano manifold (Tomoyuki Hisamoto, 2019)7Hisamoto 2019 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)3Guo 2020 Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties (Jiyuan Han and Chi Li, 2020)5Han 2020 Balanced metrics for Kähler-Ricci solitons and quantized Futaki invariants (Louis Ioos, 2021)4Ioos 2021 Lower bound of modified K-energy on a Fano manifold with degeneration for Kähler-Ricci solitons (Liang Zhang, 2021)6Zhang 2021 Optimal Degenerations of K-unstable Fano threefolds (Minghao Miao and Linsheng Wang, 2024)8Miao 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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”, “Phong Song”

  2. The Kähler-Ricci flow and optimal degenerations

    Ruadhaí Dervan and Gábor Székelyhidi

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

  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, Kähler–Ricci flow · Shared terms “Fano manifolds”

  4. Balanced metrics for Kähler-Ricci solitons and quantized Futaki invariants

    Louis Ioos

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

  5. Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties

    Jiyuan Han and Chi Li

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Kähler geometry · Shared terms “optimal degeneration”, “Fano”

  6. Lower bound of modified -energy on a Fano manifold with degeneration for Kähler-Ricci solitons

    Liang Zhang

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “degeneration”, “Fano manifolds”, “lower bound”

  7. Geometric flow, Multiplier ideal sheaves and Optimal destabilizer for a Fano manifold

    Tomoyuki Hisamoto

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

  8. Optimal Degenerations of K-unstable Fano threefolds

    Minghao Miao and Linsheng Wang

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Shared terms “optimal degeneration”, “Fano”

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 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)

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

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

  3. 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.