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Generalized almost-Kähler-Ricci solitons

Michael Albanese, Giuseppe Barbaro, Mehdi Lejmi

Abstract

We generalize Kähler-Ricci solitons to the almost-Kähler setting as the zeros of Inoue's moment map \cite{MR4017922}, 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 $4$-dimensional case. Moreover, we study the Lie algebra of holomorphic vector fields on $2n$-dimensional compact symplectic Fano manifolds admitting generalized almost-Kähler-Ricci solitons. In particular, we partially extend Matsushima's theorem \cite{MR0094478} to compact first-Chern-Einstein almost-Kähler manifolds.

Generalized almost-Kähler-Ricci solitons

Abstract

We generalize Kähler-Ricci solitons to the almost-Kähler setting as the zeros of Inoue's moment map \cite{MR4017922}, 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 \cite{MR0094478} to compact first-Chern-Einstein almost-Kähler manifolds.
Paper Structure (7 sections, 20 theorems, 125 equations)

This paper contains 7 sections, 20 theorems, 125 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Generalized almost-Kähler Ricci solitons
  4. Deforming Kähler--Ricci solitons to generalized almost-Kähler--Ricci solitons
  5. Toric Case
  6. Holomorphic vector fields on GeAKRS manifolds
  7. Proof of Lemma \ref{['lichnerowicz-AK']}

Key Result

Theorem 1

Let $(M,\omega)$ be a compact symplectic Fano manifold. Suppose that there exists $J\in AK^G_\omega$ which induces a GeAKRS with respect to a vector field $\xi$ and $\tilde{J}\in AK^G_\omega$ which induces an almost-Kähler metric of constant Chern scalar curvature. Then $\xi\equiv 0.$

Theorems & Definitions (37)

  • Theorem : Theorem \ref{['obstruction-constant-chern']}
  • Theorem : Theorem \ref{['deformations']}
  • Corollary : Corollary \ref{['existence-toric-dim4']}
  • Theorem : Theorem \ref{['killing-algebra']}
  • Corollary : Corollary \ref{['AK-Matsushima']}
  • Lemma 1
  • Definition 2
  • Proposition 3
  • Remark 4
  • Proposition 5
  • ...and 27 more