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Mabuchi Kähler solitons versus extremal Kähler metrics and beyond

Vestislav Apostolov, Abdellah Lahdili, Yasufumi Nitta

TL;DR

The paper establishes a precise equivalence between Mabuchi solitons and extremal Kähler metrics on smooth Fano manifolds by leveraging the Han–Li $v$-soliton YTD correspondence. It shows that a Mabuchi soliton exists if and only if there is an extremal Kähler metric in $2\pi c_1(X)$ with scalar curvature strictly less than $2(n+1)$, and it extends this correspondence to general $v$-solitons via a $(v,\tilde v)$-cscK framework and the weighted Mabuchi/Ding functionals. The authors relate analytic coercivity of the weighted Ding functional to geometric stability notions, including uniform relative K-stability and uniform $v$-Ding stability, across special equivariant test configurations. This yields a cohesive picture tying Mabuchi solitons to extremal metrics and to algebro-geometric stability criteria, clarifying both directions of the existence problem and providing a robust toolkit for analyzing $v$-solitons in the Fano setting.

Abstract

Using the Yau-Tian-Donaldson type correspondence for $v$-solitons established by Han-Li, we show that a smooth complex $n$-dimensional Fano variety admits a Mabuchi soliton provided it admits an extremal Kähler metric whose scalar curvature is strictly less than $2(n+1)$. Combined with previous observations by Mabuchi and Nakamura in the other direction, this gives a characterization of the existence of Mabuchi solitons in terms of the existence of extremal Kähler metrics on Fano manifolds. An extension of this correspondence to $v$-solitons is also obtained.

Mabuchi Kähler solitons versus extremal Kähler metrics and beyond

TL;DR

The paper establishes a precise equivalence between Mabuchi solitons and extremal Kähler metrics on smooth Fano manifolds by leveraging the Han–Li -soliton YTD correspondence. It shows that a Mabuchi soliton exists if and only if there is an extremal Kähler metric in with scalar curvature strictly less than , and it extends this correspondence to general -solitons via a -cscK framework and the weighted Mabuchi/Ding functionals. The authors relate analytic coercivity of the weighted Ding functional to geometric stability notions, including uniform relative K-stability and uniform -Ding stability, across special equivariant test configurations. This yields a cohesive picture tying Mabuchi solitons to extremal metrics and to algebro-geometric stability criteria, clarifying both directions of the existence problem and providing a robust toolkit for analyzing -solitons in the Fano setting.

Abstract

Using the Yau-Tian-Donaldson type correspondence for -solitons established by Han-Li, we show that a smooth complex -dimensional Fano variety admits a Mabuchi soliton provided it admits an extremal Kähler metric whose scalar curvature is strictly less than . Combined with previous observations by Mabuchi and Nakamura in the other direction, this gives a characterization of the existence of Mabuchi solitons in terms of the existence of extremal Kähler metrics on Fano manifolds. An extension of this correspondence to -solitons is also obtained.
Paper Structure (10 sections, 15 theorems, 76 equations)

This paper contains 10 sections, 15 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fano manifolds: notation and normalization
  4. Kähler--Ricci solitons
  5. $v$-solitons
  6. $v$-solitons as weighted constant scalar curvature metrics
  7. Proof of main results
  8. Mabuchi's observation for $v$-solitons
  9. Nakamura's argument for $v$-solitons
  10. The converse

Key Result

Theorem 1

[see Theorem (1,w)-YTD] Let $(X, {\mathbb T})$ be a smooth complex $n$-dimensional Fano manifold and $v$ a smooth positive weight function on $\mathrm P_X$ normalized so that $\mathit{vol}_v(X)= \mathit{vol}(X)$. Then $X$ admits a ${\mathbb T}$-invariant $v$-soliton metric in $2\pi c_1(X)$ if and on

Theorems & Definitions (28)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • Definition 1: $v$-soliton
  • Definition 2: $v$-Futaki invariant
  • Definition 3: $v$-Ding functional
  • Lemma 1
  • proof
  • Theorem 2
  • Proposition 1
  • ...and 18 more