Toric Fano manifolds that do not admit extremal Kähler metrics

Abstract

We show that there exists a toric Fano manifold of dimension $10$ that does not admit an extremal Kähler metric in the first Chern class, answering a question of Mabuchi. By taking a product with a suitable toric Fano manifold, one can also produce a toric Fano manifold of dimension $n$ admitting no extremal Kähler metric in the first Chern class for each $n\ge 11$.

Discussion on higher-dimensional cases is included. Comments welcome!