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


I just want to know given a Kähler class in $\mathbb{C}P^n\#\overline{\mathbb{C}P^n}$ ($n\geq 3$), 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 Kähler classes in $\mathbb{C}P^n\#\overline{\mathbb{C}P^n}$? If there do have, what Kähler classes in $\mathbb{C}P^n\#\overline{\mathbb{C}P^n}$ can be ruled out up to now?
Futaki invariant of course is an obstrution, but it involves the holomorphic vector field. 



A question about the existence of a constant scalar curvature metric on $\mathbb{C}P^n\#\overline{\mathbb{C}P^n}$
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".


