According the work by Wang & Zhu, on toric Fano manifolds there exist Kaehler-Ricci solitons. If Futaki=0, there also exist CSCK metrics. But if the Futaki invariant does not vanish, what about extremal metrics? Is there some counterexample?

## 2 Answers

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.

Some additional comment:

Let $(X,J,\omega)$ be Fano Kahler manifold with $\omega_0\in [\omega_0]=\kappa$, then by $\partial\bar\partial$-lemma any other Kahler 2-form can be written as $\omega_0+\sqrt{-1}\partial\bar\partial \varphi$. Now the set of Kahler metrics in $[\omega_0]$ can be identified with $$\mathcal H=\{\varphi \in C^\infty(X,\mathbb R)|\omega_0+\sqrt{-1}\partial\bar\partial \varphi>0\}/\mathbb R$$. Note that $T_\varphi\mathcal H=C^\infty(X,\mathbb R)/\mathbb R$

Now, consider the Calabi- functional(which is sort of special case of Yang-Mills functional)

If we denote $S(\omega_\varphi)$ denotes to be the scalar curvature of the metric $\omega_\varphi$ and $\underline S$ be the average scalar curvature

$$\mathcal C a:\mathcal H\to \mathbb R$$ $$\varphi\to \mathcal Ca(\varphi)=\int_X (S(\omega_\varphi)-\underline S)^2\frac{\omega_\varphi^n}{n!}$$

Now if you take $\psi\in T_\varphi\mathcal H$ , then we can write the variation of Calabi functional via using Lichnerowicz operator $\mathcal D_\varphi=\overline \partial\nabla$ as follows

$$(d\mathcal Ca(\varphi))\psi=2\int_X (\mathcal D_\varphi^*\mathcal D_\varphi S(\omega_\varphi)).\psi\frac{\omega_\varphi^n}{n!}$$ where $\mathcal D_\varphi^*$ is the $L^2(X,\omega_\varphi)$-adjoint of $\mathcal D_\varphi$. As an exercise integrating by part we have $\ker \mathcal D_\varphi^*\mathcal D_\varphi =\ker \mathcal D_\varphi$

This means that $\omega_\varphi$ is extremal metric (which means the critical points of Calabi functional) if the gradient of scalar curvature $S(\omega_\varphi)$ define a vector field .

About your question this recent paper https://arxiv.org/pdf/1610.06865.pdf