Let $X$ be a nonsingular projective variety over $\mathbb{C}$, then it has a complex manifold structure, denoted by $X_{an}$. Then by Serre's theorem:
Let $X$ be a projective variety over $\mathbb{C}$, then there is an equivalence of categories from the category of coherent sheaves on $X$ to the category of coherent analytic sheaves on $X_h$. Furthermore, for every coherent sheaf $F$ on $X$, (recall every projective algebraic variety $X$, we can view it as an analytic variety, denoted as $X_h$) we have: $$H^i(X,F)\cong H^i((X_h),F_h)$$
It follows that $\Gamma(X,\mathcal{O}_X)\cong \Gamma(X,\mathcal{O}_{X_{an}})$, where $\mathcal{O}_X$ is the structure sheaf on variety $X$ and $\mathcal{O}_{X_{an}}$ is the structure sheaf on complex manifold. My question is if there is an alternative way to see $\Gamma(X,\mathcal{O}_X)\cong \Gamma(X_{an},\mathcal{O}_{X_{an}})$ without using Serre's theorem, or is this obvious to see $\Gamma(X,\mathcal{O}_X)\cong \Gamma(X_{an},\mathcal{O}_{X_{an}})$?