Paper · Feb 6, 2024

Generalized almost-Kähler-Ricci solitons

Michael Albanese, Giuseppe Barbaro, Mehdi Lejmi

math.DGarXiv:2402.03996
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As on arXiv

Abstract

We generalize Kähler-Ricci solitons to the almost-Kähler setting as the zeros of Inoue's moment map, and show that their existence is an obstruction to the existence of first-Chern-Einstein almost-Kähler metrics on compact symplectic Fano manifolds. We prove deformation results of such metrics in the -dimensional case. Moreover, we study the Lie algebra of holomorphic vector fields on -dimensional compact symplectic Fano manifolds admitting generalized almost-Kähler-Ricci solitons. In particular, we partially extend Matsushima's theorem to compact first-Chern-Einstein almost-Kähler manifolds.

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Submitted
Feb 6, 2024
Subjects
math.DG, math.CV
Ricci solitonsKähler–Ricci solitonsKähler geometry

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  1. Toric geometry of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al.

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author Giuseppe Barbaro · Shared terms “symplectic”

  2. Kählerity of complete almost-Kähler gradient shrinking Ricci solitons

    Junming Xie

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

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

  4. Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton

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    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “vector field”

  5. Kähler-Ricci flow on -spherical Fano manifolds

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “Fano manifold”, “Lie algebras”

  6. Special Ricci-Hessian equations on Kähler manifolds

    Andrzej Derdzinski and Paolo Piccione

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “Kähler metrics”, “Einstein”

  7. Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds

    Liding Huang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “Fano manifold”, “symplectic”, “Kähler metrics”

  8. Variational structure and uniqueness of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons

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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… 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.