In this talk, we are concerned with the relation between two kinds of canonical Kähler metrics on Fano manifolds, the Calabi's extremal Kähler metrics and the Mabuchi solitons (or generalized Kähler-Einstein metrics in the literature). These are both generalizations of the concept of Kähler-Einstein metrics.
Mabuchi showed that the existence of Mabuchi solitons implies that of extremal Kähler metrics representing the first Chern class. It is also known that the converse is true for Fano manifolds of dimension up to two.
Based on the above, we present examples of Fano manifolds in ALL dimensions greater than two which admit extremal Kähler metrics in every Kähler class, but do not admit Mabuchi solitons.
A key ingredient is the Mabuchi constant, a holomorphic invariant of a Fano manifold, and its product formula.
This talk is based on a joint work with Shunsuke Saito.