GAGA pullback of sheaf of Kähler differential. Basically the question is here Analytification of algebraic differential forms for smooth complex projective scheme of finite type $X$. Let $h: X^{an} \rightarrow X$ be the analytification morphism. I would love to see a more detailed proof why the pullback of $\Omega_{X/\mathbb{C}}$, sheaf of Kähler differentials of $X$, is $\Omega^{an}$, the sheaf of holomorphic 1-forms of $X^{an}$. I think we just need to construct a canonical morphism $h^*\Omega_{X/\mathbb{C}} \rightarrow \Omega^{an}$ and prove that it is injective or surjective, then the isomorphism follows from the fact that both are locally free of the same rank.
Thank you in advance.
 A: Consider smooth scheme $X$ over $\mathbb{C}$. Fix a closed point $x\in X$ and an open affine neighborhood $U$ of $x$ in $X$. Let $\mathbb{C}[U]$ be the coordinate ring of $U$. An ideal $\mathfrak{m}_x$ of $x$ in $C[U]$ is maximal and it admits a set of generators $z_1,...,z_n$ such that their classes in $\mathfrak{m}_x/\mathfrak{m}^2_x$ are linearly independent. We define a morphism of $\mathbb{C}$-algebras
$$\mathbb{C}[t_1,...,t_n]\rightarrow C[U]$$
such that $t_i \mapsto z_i$ for every $i=1,2,...,n$. This morphism corresponds to a morphism of $\mathbb{C}$-schemes
$$\phi:U\rightarrow \mathbb{A}^n_{\mathbb{C}}$$
given by functions $z_1,...,z_n$. Since $z_1,...,z_n$  form a minimal set of generators of the maximal ideal $\mathfrak{m}_x$ and $U$ is smooth, we derive that $\phi$ is an étale morphism on some open affine neighborhood $V$ of $x$ in $U$ (this is just an invocation of the fact that étale locus is Zariski open). Hence we obtain an étale morphism $\phi_{\mid V}:V\rightarrow \mathbb{A}^n_{\mathbb{C}}$.
Let $i:V^{\mathrm{an}}\rightarrow V$ be the canonical morphism of $\mathbb{C}$-ringed spaces.
Now we make two observations

*

*The sheaf $\Omega_V$ of algebraic differential $1$-forms is a free sheaf generated by $dz_1$,...,$dz_n$ (this follows from the fact that $\phi_{\mid V}$ is étale).


*The analytification ${\phi_{\mid V}}^{\mathrm{an}}:V^{\mathrm{an}}\rightarrow \mathbb{C}^n$ is locally biholomorphic (because it is analytification of an étale morphism) and given by $i^{\#}(z_1),...,i^{\#}(z_n)$ ($z_1,...,z_n$ viewed as holomorphic functions on $V^{\mathrm{an}}$). Hence $di^{\#}(z_1),...,di^{\#}(z_n)$ generate freely the sheaf $\Omega_{V^{\mathrm{an}}}$ of holomorphic differential forms on $V^{\mathrm{an}}$
Observations 1) and 2) imply that the analytification
$$(\Omega_V)^{\mathrm{an}} =i^*\Omega_V = \left(i^{-1}\Omega_V\right)\otimes_{i^{-1}\mathcal{O}_V} \mathcal{O}_{V^\mathrm{an}}$$
of algebraic $1$-forms is a free $\mathcal{O}_{V^\mathrm{an}}$-sheaf generated by $i^*(dz_1),...,i^*(dz_n)$ and the sheaf of holomorphic $1$-forms $\Omega_{V^{\mathrm{an}}}$ is free over $\mathcal{O}_{V^\mathrm{an}}$ and generated by $di^{\#}(z_1),...,di^{\#}(z_n)$. Since the canonical morphism
$$i^*\Omega_V\rightarrow \Omega_{V^{\mathrm{an}}}$$
is given by $i^*(df)\mapsto di^{\#}(f)$, we derive that $i^*\Omega_V\rightarrow \Omega_{V^{\mathrm{an}}}$ is an isomorphism of sheaves of $\mathcal{O}_{V^{\mathrm{an}}}$-modules. Since this morphism of sheaves is the restriction to $V^{\mathrm{an}}$ of a morphism
$$j^*\Omega_X\rightarrow \Omega_{X^{\mathrm{an}}}$$
induced by the analytification morphism $j:X^{\mathrm{an}}\rightarrow X$, we derive that $j^*\Omega_X\rightarrow \Omega_{X^{\mathrm{an}}}$ is an isomorphism.
