Extremal Kähler-Ricci solitons on Fano homogeneous toric bundles (Q6621297)

A central problem in Kähler geometry is to determine whether a Kähler manifold admits a canonical Kähler metric in a given Kähler class. One natural such canonical metric equation is the Kähler-Einstein equation, and there exist many generalisations, such as constant scalar curvature or, more generally, extremal Kähler metrics, and Kähler-Ricci solitons. One can ask whether a Kähler metric can satisfy several of these equations simultaneously. \textit{S. Calamai} and \textit{D. Petrecca} [Proc. Am. Math. Soc. 144, No. 2, 813--821 (2016; Zbl 1337.53053)] studied this on compact Fano manifolds for the extremal and Kähler-Ricci soliton equations. Kähler-Einstein metrics satisfy both of these equations, and the question they posed is whether there exists such metrics which are not Kähler-Einstein. They called these metrics extremal Kähler-Ricci solitons, and showed both that extremal Kähler-Ricci solitons with positive holomorphic sectional curvature and on toric manifolds necessarily are Kähler-Einstein.\N\NThe article under review is a short note that addresses this question in the context of homogeneous toric bundles, which are fibrations over a generalized flag manifold \(G/K\) with toric fibres. The main result is that an extremal Kähler-Ricci soliton on a Fano homogeneous toric bundle is necessarily Kähler-Einstein, generalising the results of Calamai and Petrecca from toric manifolds to homogeneous toric bundles. The proof goes via manipulating the symplectic potential for the toric fibre metrics in action-angle coordinates, ultimately showing that the soliton vector field of the Kähler-Ricci soliton has to vanish if the metric is also extremal.