Generalized almost-Kähler-Ricci solitons ☆

We generalize K & auml;hler-Ricci solitons to the almost-K & auml;hler setting as the zeros of Inoue's moment map [25], and show that their existence is an obstruction to the existence of first-Chern-Einstein almost-K & auml;hler 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-K & auml;hler-Ricci solitons. In particular, we partially extend Matsushima's theorem [41] to compact first-Chern-Einstein almost-K & auml;hler manifolds. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data