A carefully built 10-dimensional toric Fano manifold satisfies a relative K-instability criterion, so it cannot carry an extremal Kahler metric in c1.
Calabi, Extremal K\"ahler metrics , Seminar on Differential Geometry, Ann
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Toric Fano manifolds that do not admit extremal K\"ahler metrics
A carefully built 10-dimensional toric Fano manifold satisfies a relative K-instability criterion, so it cannot carry an extremal Kahler metric in c1.