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According the work by Wang & Zhu, on toric Fano manifolds there exists Kahler-Ricci solitons, if futaki=0 there exist CSCK metric, but if futaki invariant is not vanished, what about extremal metrics? Is there some counterexample?

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up vote 6 down vote accepted

Here is an example where the Futaki invariant does not vanish, but with extremal metrics. The Futaki invariant on $Bl_p\mathbb{P}^2$, the blow up of $\mathbb{P}^2$, which is toric, does not vanish (Canonical Metrics in Kaehler geometry by Tian, Example 3,10). On the other hand, $Bl_p\mathbb{P}^2$ admits an extremal metric in every Kaehler class (I believe this is in Calabi's Extremal Kaehler Metrics). Is this what you are looking for?

Conjecturally, the existence of an extremal metric is equivalent to relative K-stability, see the work of Székelyhidi. The usual K-stability involved in the Yau-Tian-Donaldson conjecture is a generalisation of the classical Futaki invariant, and relative K-stability is an adaptation of this to the extremal setting. Székelyhidi (in his thesis) has shown that the existence of an extremal metric implies relative K-semistability, so this forms an obstruction. For toric surfaces, this was studied by Bohui Chen, An-Min Li, Li Sheng in "Extremal metrics on Toric Surfaces". In particular, they show relative K-stability is equivalent to the existence of an extremal metric on toric surfaces. I guess not much is known in higher dimensions.

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