Paper · Apr 4, 2022

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

Yasuhiro Nakagawa, Satoshi Nakamura

math.DGarXiv:2204.01217v3
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Abstract

In this article we focus on multiplier Hermitian-Einstein metrics introduced by Mabuchi which include Kähler-Einstein metrics, Kähler-Ricci solitons and Mabuchi solitons as special cases. We also focus on KSM-manifolds, which are introduced by the first author as toric bundles, to establish a criterion for the existence of multiplier Hermitian-Einstein metrics in terms of KSM-data. An explicit example for a KSM-manifold admitting a family of multiplier Hermitian-Einstein metrics is constructed by using a continuous path connecting a Kähler-Ricci soliton and a Mabuchi soliton.

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Submitted
Apr 4, 2022
Last revised
Aug 10, 2022 (v3)
Subjects
math.DG

Comments: 31 pages, final version, to appear in Tohoku Mathematical Journal

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryKähler–Einstein & K-stability

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  1. Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein

    Yasufumi Nitta

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

  2. Deformations of Fano manifolds with weighted solitons

    Akito Futaki

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

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

    Charles Cifarelli, Ronan J. Conlon and Alix Deruelle

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

  4. The Miyaoka-Yau inequality and the delta invariant for Fano varieties II -soliton case-

    Tomoyuki Hisamoto and Masataka Iwai

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

  5. Log del Pezzo -surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics

    Daniel Hättig, Jürgen Hausen and Hendrik Süß

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

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

    Liding Huang and Xiaohua Zhu

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

  7. estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation

    Wangjian Jian and Yalong Shi

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Shared terms “Fano manifold”

  8. The Kähler-Ricci soliton on bounded pseudoconvex domains

    Zehao Sha

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics

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. 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.