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 group 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".

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".