[proofplan] Smoothness of a map between manifolds is local in charts, so we fix an arbitrary point of $X$ and work near it. In holomorphic charts around the point and its image, the map $F$ is represented by a holomorphic map between open subsets of complex Euclidean spaces. The standard [regularity theorem](/theorems/2750) from several complex variables says that such a holomorphic coordinate representation is $C^\infty$ as a real map. Since the real smooth charts underlying the complex manifolds are precisely the realifications of the holomorphic charts, this verifies the coordinate criterion for smoothness. [/proofplan] [step:Express the map in holomorphic coordinates around an arbitrary point] Let $p\in X$ be arbitrary. Since $F:X\to Y$ is holomorphic, there exist holomorphic charts $(U,\varphi)$ on $X$ around $p$ and $(V,\psi)$ on $Y$ around $F(p)$ such that $F(U)\subset V$ and the coordinate representation \begin{align*} f:=\psi\circ F\circ\varphi^{-1}:\varphi(U)\to\psi(V) \end{align*} is holomorphic. Here $\varphi(U)\subset\mathbb{C}^n$ and $\psi(V)\subset\mathbb{C}^m$ are open sets, where $n=\dim_{\mathbb{C}}X$ and $m=\dim_{\mathbb{C}}Y$. [guided] We must prove smoothness at every point of $X$, so we begin by fixing an arbitrary point $p\in X$. The definition of a holomorphic map between complex manifolds is local in holomorphic charts: around $p$ and around $F(p)$, the map can be written as a holomorphic map between open subsets of complex Euclidean spaces. Thus choose a holomorphic chart $(U,\varphi)$ on $X$ with $p\in U$ and a holomorphic chart $(V,\psi)$ on $Y$ with $F(p)\in V$ such that $F(U)\subset V$. The corresponding coordinate representation is the map \begin{align*} f:=\psi\circ F\circ\varphi^{-1}:\varphi(U)\to\psi(V). \end{align*} By the defining property of holomorphic maps between complex manifolds, this map $f$ is holomorphic. The sets $\varphi(U)\subset\mathbb{C}^n$ and $\psi(V)\subset\mathbb{C}^m$ are open because $\varphi$ and $\psi$ are charts, where $n=\dim_{\mathbb{C}}X$ and $m=\dim_{\mathbb{C}}Y$. [/guided] [/step] [step:Use complex differentiability to obtain real smoothness of the coordinate representation] View $\varphi(U)\subset\mathbb{C}^n$ and $\psi(V)\subset\mathbb{C}^m$ as open subsets of $\mathbb{R}^{2n}$ and $\mathbb{R}^{2m}$ through the standard identification \begin{align*} (z_1,\dots,z_n)\mapsto (\operatorname{Re}z_1,\operatorname{Im}z_1,\dots,\operatorname{Re}z_n,\operatorname{Im}z_n). \end{align*} By the standard theorem from several complex variables that holomorphic maps on open subsets of $\mathbb{C}^n$ are $C^\infty$ as real maps, the coordinate representation $f:\varphi(U)\to\psi(V)$ is smooth as a map from an open subset of $\mathbb{R}^{2n}$ to an open subset of $\mathbb{R}^{2m}$. Equivalently, each real component of $f$ is $C^\infty$. [/step] [step:Translate the coordinate computation back to the underlying smooth manifolds] The canonical smooth structure on a [complex manifold](/page/Complex%20Manifold) is defined by taking every holomorphic chart as a smooth chart after the standard real identification $\mathbb{C}^k\cong\mathbb{R}^{2k}$. Therefore $(U,\varphi)$ and $(V,\psi)$ are smooth charts for the underlying smooth manifolds of $X$ and $Y$. The coordinate expression of $F$ in these smooth charts is exactly \begin{align*} \psi\circ F\circ\varphi^{-1}=f. \end{align*} The preceding step shows that this coordinate expression is smooth. Since $p\in X$ was arbitrary, the coordinate criterion for smooth maps between smooth manifolds gives that $F:X\to Y$ is smooth as a map between the underlying smooth manifolds. This proves the theorem. [/step]