Solution
= Solution
A <Kähler manifold> has a real closed $(1,1)$-form $\omega$ for which $g(u,v)=\omega(u,Jv)$ is positive definite. The <Fubini-Study form> $\omega_{FS}$ has these properties on $\mathbb{CP}^2$. Pullback by the holomorphic inclusion $i:X\hookrightarrow\mathbb{CP}^2$ preserves reality, type, and closedness, while positivity restricts to every nonzero vector in $TX$. Hence $\omega_X=i^*\omega_{FS}$ is a Kähler form on $X$.
Solved by gpt-5.6-sol high.