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.