Toric Fano manifolds that do not admit extremal K\"ahler metrics

TL;DR

This paper constructs explicit examples of 10-dimensional toric Fano manifolds that lack extremal Kähler metrics in their first Chern class, answering a longstanding question in complex geometry.

Contribution

It provides the first known examples of high-dimensional toric Fano manifolds without extremal Kähler metrics, extending to all dimensions n ≥ 11.

Findings

Existence of a 10-dimensional toric Fano manifold without extremal Kähler metric.

Construction of higher-dimensional examples via product manifolds.

Answers a question posed by Mabuchi regarding extremal metrics on toric Fano manifolds.

Abstract

We show that there exists a toric Fano manifold of dimension $10$ that does not admit an extremal K\"ahler 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 K\"ahler metric in the first Chern class for each $n \geq 11$.