Proposition 5.2.4A683A13C69D9

Proof. (1) The real derivative of $F$ at $a$ is the complex-linear map given by the invertible matrix $(\partial _{z_ k}F_ j(a))$, so it is invertible. By the real inverse function theorem there are open neighborhoods $U'$ of $a$ and $V'$ of $F(a)$ such that $F$ maps $U'$ bijectively onto $V'$, the inverse map $G\colon V'\to U'$ is smooth, and $\mathrm dG_ w=(\mathrm dF_{G(w)})^{-1}$ for $w\in V'$. Since $F$ is holomorphic, $\mathrm dF_{G(w)}$ is complex linear, and hence so is its inverse. Thus the real derivative of $G$ is complex linear at every point, that is, $G$ is holomorphic.

(2) If $r=0$, the identity map has the required property. Otherwise some $r\times r$ minor of the matrix $(\partial _{z_ k}f_ j(a))$ is nonzero; let $k_1<\cdots <k_{m-r}$ be the indices of the columns not used by this minor. The map $\Phi =(f_1,\ldots ,f_ r,z_{k_1},\ldots ,z_{k_{m-r}})$ is holomorphic on $U$. The last $m-r$ rows of its complex Jacobian matrix at $a$ are the unit row vectors with entry $1$ in the columns $k_1,\ldots ,k_{m-r}$, so, expanding the determinant along these rows, we see that it equals the chosen minor up to sign; hence it is nonzero. By (1), $\Phi $ maps an open neighborhood of $a$ bijectively onto an open subset of $\mathbb C^ m$, with holomorphic inverse.

(3) The rank of the complex Jacobian matrix does not depend on the chart, by the chain rule, since coordinate changes have invertible complex Jacobian matrices. Let $\kappa \colon U\to \kappa (U)\subset \mathbb C^ m$ be a holomorphic chart around $a$ on which the $f_ j$ are defined. By (2), applied to the functions $f_ j\circ \kappa ^{-1}$ at $\kappa (a)$, there is a map $\Phi $ from an open neighborhood $V\subset \kappa (U)$ of $\kappa (a)$ onto an open subset of $\mathbb C^ m$, biholomorphic and with first components $f_ j\circ \kappa ^{-1}$. Then $\Phi \circ \kappa $ is a homeomorphism of $\kappa ^{-1}(V)$ onto an open subset of $\mathbb C^ m$ whose first $r$ components are $f_1,\ldots ,f_ r$. For every chart $\lambda $ of the atlas, the coordinate changes $(\Phi \circ \kappa )\circ \lambda ^{-1}=\Phi \circ (\kappa \circ \lambda ^{-1})$ and $\lambda \circ (\Phi \circ \kappa )^{-1}=(\lambda \circ \kappa ^{-1})\circ \Phi ^{-1}$ are compositions of holomorphic maps, so $\Phi \circ \kappa $ is a holomorphic chart. $\square$