# a question about the existence of constant scalar curvature metric on $\mathbb{C}P^n\sharp\bar{\mathbb{C}P^n}$

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