Kähler metric on the projective space

Question

"Is there a Kähler metric on the complex projective space $\mathbb {P} ^n(\mathbb {C} ) $ different from the Fubini-Study metric?

Answer 1

Every Kähler manifold admits an infinite dimensional family of Kähler metrics, by adding a Kähler potential with small enough second derivatives. One can even do this in any open set, using compactly supported Kähler potentials. Note that this might not alter the Kähler class. Just rescaling any given Kähler metric already gives another one, which (on any compact manifold) has a different Kähler class. Kähler classes sit as an open cone in the cohomology of any compact complex manifold. For a detailed description of the Kähler cone, see Demailly, Numerical characterization of the Kähler cone of a compact Kähler manifold, Annals of Mathematics, 159 (2004), 1247–1274. For complex projective space, since $h^{1,1}(\mathbb{CP}^n)=1$, we see that the Kähler cone consists of the positive rescalings of the Kähler class of the usual Fubini-Study metric.