Let $H = \{z_0 = 0\}$ be a hyperplane in $\Bbb{CP}^n$, I know that the local defining equation will be on $U_0 = \{z_0 \ne 0\}$ is $1$ , and on $U_i = \{z_i \ne 0\}$ for $i>1$ is $z_0/z_i$ (as quotient is well defined on the projective space).
I have a silly question does the hyperplane has a global smooth(holomorphic) defining function? (that is some $f$ defined on $\Bbb{CP}^n$ which is defined to be smooth (holomorphic) such that $H = f^{-1}(0)$) is quite tempting to claim that $z_0$ is the global defining function , but it's not well defined on $\Bbb{CP}^n$.