If $X$ is $\mathbb{C}P^n$ as a projective variety, and $X_h$ is the corresponding analytic structure. Then do we have an isomorphism $\mathcal{O}_X\cong \mathcal{O}_{X_{h}}$ for the structure sheaves?
More generally, do we have $\mathcal{O}_X\cong \mathcal{O}_{X_{h}}$ if $X$ is a nonsingular projective variety?