Mabuchi Kähler solitons versus extremal Kähler metrics and beyond

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.