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We know that Calabi constructed some extremal metrics on $\mathbb{C}P^n\sharp\bar{\mathbb{C}P^n}$ which are not constant scalar curvature ones.

I just want to know given a K\"{a}hler class in $\mathbb{C}P^n\sharp\bar{\mathbb{C}P^n}$ ($n\geq 3$), is there any criterion to detemine whether or not there contains a constant scalar curvature metric? Or are there some known obstructions which can be used to rule out the existence of csc metrics in some K\"{a}hler classes in $\mathbb{C}P^n\sharp\bar{\mathbb{C}P^n}$? If there do have, what K\"{a}hler classes in $\mathbb{C}P^n\sharp\bar{\mathbb{C}P^n}$? can be ruled out up to now? Futaki invariant of course is an obstrution, but it involves the holomorphic vector field.

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The Futaki invariant is indeed an obstruction. Your manifold is toric, so for torus-invariant Kahler classes it can be computed explicitly from the moment polytope (see e.g. arxiv.org/pdf/0803.0985 ) –  YangMills May 1 '12 at 13:43
    
Thanks for your reply. What do torus-invariant Kahler classes mean? Given a Kahler class on a toric manifold, it doesnot change under this toric action, isn't it? OK. Maybe I should ask my question more explicitly. Let $x$ and $y$ be the two generators of $H^2(CP^n\sharp\bar{CP^n})$ corresponding to the $CP^n$ and $\bar{CP^n}$ respectively. We assume $\int x^n=-\int y^n=1$. Now any Kahler class should be written in the form $ax+by$ $(a^n-b^n>0)$. My question is, for what pairs $(a,b)$(a^n-b^n)$, the correponding Kahler classes don't contain csc metrics. I want to know the concrete results. –  Ping May 2 '12 at 0:33
    
Sorry. In the last sentence, $(a,b)$(a^n-b^n)& should be $(a,b)$ $(a^n-b^n>0)$ –  Ping May 2 '12 at 0:36
    
On a compact toric manifold there are in general non-torus invariant Kahler classes. –  YangMills May 2 '12 at 15:09
    
Thanks. But I just want to know what kinds of $(a,b)$ cannot admit CSC metric:-) –  Ping May 2 '12 at 23:16
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1 Answer 1

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There are no constant scalar curvature Kähler metrics on $M$, the blowup of $\mathbb{CP}^n$ at one point in any Kähler class and for any $n>1$.

This is because of the Lichnerowicz-Matsushima obstruction, which in this case says that if any such metric existed then $Aut^0(M)$ the connected component of the identity of the automorphism group of $M$ would be reductive (this is because $M$ is Fano so all holomorphic vector fields have a zero somewhere).

But $Aut^0(M)$ is readily seen to be isomorphic to the subgroup of $PGL(n+1,\mathbb{C})$ of matrices (modulo multiples of the identity) with arbitrary entries except for the first column which looks like $(*,0,0,\dots,0)$ where * is any nonzero complex number. This group is not reductive, so $M$ does not admit any constant scalar curvature Kähler metric in any class.

There are of course extremal Kähler metrics in some classes, which have nonconstant scalar curvature, the first example was constructed by Calabi in 1982 in his paper "Extremal Kähler metrics".

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Yes, you are right:-) This is the most important obstruction before Futaki's integral invariant, which I suddenly forgot:-) We can also use it to show that the blowup of $\mathbb{CP}^n$ at two genric points also cannot any csc metric. –  Ping May 8 '12 at 0:55
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