Paper · Jun 1, 2020

On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations

Jiyuan Han, Chi Li

math.DGarXiv:2006.00903v3
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Abstract

Let be a log variety with an effective holomorphic torus action, and be a closed positive -current. For any smooth positive function defined on the moment polytope of the torus action, we study the Monge-Ampère equations that correspond to generalized and twisted Kähler-Ricci -solitons. We prove a version of Yau-Tian-Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform -twisted -Ding-stability. When is a current associated to a torus invariant linear system, we further show that equivariant special test configurations suffice for testing the stability. Our results allow arbitrary klt singularities and generalize most of previous results on (uniform) YTD conjecture for (twisted) Kähler-Ricci/Mabuchi solitons or Kähler-Einstein metrics.

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Submitted
Jun 1, 2020
Last revised
Oct 20, 2020 (v3)
Subjects
math.DG, math.AG

Comments: 58 pages

Ricci solitonsKähler–Ricci solitonsEinstein metricsSingularitiesKähler geometryStability & uniquenessKähler–Einstein & K-stabilityAnalytic minimal model program

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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 K-Stability of Fano spherical varieties (Thibaut Delcroix, 2016)1Delcroix 2016 Coupled Kähler-Ricci solitons on toric Fano manifolds (Jakob Hultgren, 2017)5Hultgren 2017 Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons (Yi Yao, 2019)2Yao 2019 Tian's partial C⁰-estimate implies Hamilton-Tian's conjecture (Feng Wang and Xiaohua Zhu, 2020)7Wang 2020 Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties (Jiyuan Han and Chi Li, 2020)4Han 2020 Notes on weighted Kähler-Ricci solitons and application to Ricci-flat Kähler cone metrics (Chi Li, 2021)3Li 2021 On the Existence of Weighted-cscK Metrics (Jiyuan Han and Yaxiong Liu, 2024)6Han 2024 On Kähler-Einstein Currents (Yifan Chen, Shih-Kai Chiu, Max Hallgren et al., 2025)8Chen 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. K-Stability of Fano spherical varieties

    Thibaut Delcroix

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Tian Donaldson”, “Yau Tian”, “Donaldson conjecture”

  2. Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons

    Yi Yao

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Ding stability”, “Mabuchi solitons”, “Tian Donaldson”

  3. Notes on weighted Kähler-Ricci solitons and application to Ricci-flat Kähler cone metrics

    Chi Li

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Same author Chi Li · Shared terms “YTD”, “Tian Donaldson”, “Yau Tian”

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

    Jiyuan Han and Chi Li

    Topics Kähler–Einstein & K-stability, Kähler geometry, Stability & uniqueness · Same authors Jiyuan Han, Chi Li · Shared terms “test configurations”

  5. Coupled Kähler-Ricci solitons on toric Fano manifolds

    Jakob Hultgren

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Tian Donaldson”, “Yau Tian”, “Donaldson conjecture”

  6. On the Existence of Weighted-cscK Metrics

    Jiyuan Han and Yaxiong Liu

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler geometry · Same author Jiyuan Han · Shared terms “Mabuchi”

  7. Tian's partial -estimate implies Hamilton-Tian's conjecture

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Shared terms “Tian Donaldson”, “Yau Tian”, “Donaldson conjecture”

  8. On Kähler-Einstein Currents

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

    Topics Analytic minimal model program, Kähler–Ricci solitons, Kähler–Einstein & K-stability

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

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