Paper · Dec 18, 2018

Coupled complex Monge-Ampère equations on Fano horosymmetric manifolds

Thibaut Delcroix, Jakob Hultgren

math.DGarXiv:1812.07218v2
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Abstract

We give necessary and sufficient conditions for existence of solutions to a general system of complex Monge-Ampère equations on Fano horosymmetric manifolds. In particular, we get necessary and sufficient conditions for existence of coupled Kähler-Ricci solitons, Mabuchi metrics and twisted Kähler-Einstein metrics in terms of combinatorial data of the manifold.

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Submitted
Dec 18, 2018
Last revised
May 6, 2022 (v2)
Subjects
math.DG
Journal
J Math Pures Appl (9) 153, 281--315 (2021)
DOI
10.1016/j.matpur.2020.12.002

Comments: accepted version

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler 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 K-Stability of Fano spherical varieties (Thibaut Delcroix, 2016)4Delcroix 2016 Coupled Kähler-Ricci solitons on toric Fano manifolds (Jakob Hultgren, 2017)2Hultgren 2017 Coupled Sasaki-Ricci solitons (Akito Futaki and Yingying Zhang, 2018)8Futaki 2018 Examples of K-unstable Fano manifolds (Thibaut Delcroix, 2019)1Delcroix 2019 Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type (Yasuhiro Nakagawa and Satoshi Nakamura, 2022)7Nakagawa 2022 An Aubin continuity path for shrinking gradient Kähler-Ricci solitons (Charles Cifarelli, Ronan J. Conlon and Alix Deruelle, 2022)5Cifarelli 2022 Horosymmetric limits of Kähler-Ricci flow on Fano G-manifolds (Gang Tian and Xiaohua Zhu, 2022)3Tian 2022 Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces (Ye-Won Luke Cho and Young-Jun Choi, 2024)6Cho 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Examples of K-unstable Fano manifolds

    Thibaut Delcroix

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Same author Thibaut Delcroix · Shared terms “horosymmetric manifolds”, “coupled Kähler”, “Fano”

  2. Coupled Kähler-Ricci solitons on toric Fano manifolds

    Jakob Hultgren

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Same author Jakob Hultgren · Shared terms “coupled Kähler”, “Fano”, “sufficient conditions”

  3. Horosymmetric limits of Kähler-Ricci flow on Fano -manifolds

    Gang Tian and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “Fano horosymmetric”, “horosymmetric manifolds”

  4. K-Stability of Fano spherical varieties

    Thibaut Delcroix

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Same author Thibaut Delcroix · Shared terms “combinatorial data”, “Fano”

  5. An Aubin continuity path for shrinking gradient Kähler-Ricci solitons

    Charles Cifarelli, Ronan J. Conlon and Alix Deruelle

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “complex Monge”, “Ampère equation”, “Monge Ampère”

  6. Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces

    Ye-Won Luke Cho and Young-Jun Choi

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Shared terms “complex Monge”, “Ampère equation”, “Monge Ampère”

  7. 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”, “Mabuchi”

  8. Coupled Sasaki-Ricci solitons

    Akito Futaki and Yingying Zhang

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “coupled 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) 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.