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The first Hirzebruch surface $F_1$ does not admit any Kaehler metric of constant scalar curvature. Yet it admits an extremal metric representing each Kaehler class as shown by Calabi. The two points blow-up of $\mathbb{CP}^2$ behaves similarly.

Do you know any other example of Fano manifold such that:

  1. The anti-canonical class does not contain any Kaehler-Einstein metric.
  2. The anti-canonical class contains an extremal metric.

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I'm not 100% sure what you're looking for because of the wording of your question. Do you want a Kähler manifold for which every Kähler class has an extremal representative, but no constant scalar curvature representative? Or examples of manifolds with just one such Kähler class?

If it's the latter, then note that if there is a metric of constant scalar curvature in the class there can be no genuinely extremal metrics (since the Futaki invariant must vanish). (If I misunderstood your question and you already knew this then sorry for teaching you to suck eggs!) For plenty of concrete examples of extremal metrics you could look in the article of Arezzo-Pacard-Singer for blow-ups:

http://arxiv.org/abs/math/0701028

Alternatively, you can try the paper of Chen-Li-Sheng, which (building on work of Donaldson in the case of constant scalar curvature) settles the problem completely for toric surfaces (of which $F_1$ is an example):

http://arxiv.org/abs/1008.2607

From here you might be able to find more toric surfaces which have the property that all Kähler classes have extremal but not constant scalar curvature representatives. It will come down to some calculations involving polygons, but they could well be very difficult. Perhaps trying them in the case of $F_1$ would show how to find another example.

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  • $\begingroup$ It's just dawning on me which Yann you probably are. Salut! In which case you can skip the first two paragraphs of my answer, and you'd probably already thought of the second two yourself... $\endgroup$
    – Joel Fine
    Jun 21, 2011 at 16:17
  • $\begingroup$ Thanks Joel ! My question was indeed a bit too vague contrarily to your answer. In fact, I am specifically interested in Fano manifolds, and Kaehler class close to the canonical one; this is why I mentioned $F_1$. I am going to modify my question accordingly. $\endgroup$
    – Yann
    Jun 21, 2011 at 16:44
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In fact, I am asking the question because I just proved that there are certain deformations of the Mukai-Umemura $3$-fold that belong to Tian's family and admit extremal metrics in the anti-canonical Kaehler classes. However they do not admit any Kaehler-Einstein metric according to the celebrated result of Tian. The remarkable thing is that these examples are arbitrarily small complex deformations of the Mukai-Umemura $3$-fold which does admit a Kaehler-Einstein metric by Donaldson's computation of the $\alpha$-invariant.

I would like to know how much this example of phenomenon is original, and if there are others of this type.

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  • $\begingroup$ That's interesting, but sounds impossible. If the deformation preserves a positive dimensional subgroup of the automorphism group of the mukai-Umemura 3-fold, then the Futaki invariant has to vanish on the subgroup. This follows from an argument in arxiv.org/abs/hep-th/0603021 and arxiv.org/abs/math/0607586 that the Futaki invariant vanishes on a subgroup iff a volume functional is minimized on a "moment cone". The symplectic structure on the cone is not changed, thus the volume functional remains minimized. $\endgroup$
    – Craig
    Mar 17, 2013 at 3:51

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