Definition:Complex Analytic Differentiable Structure

Definition
Let $M$ be a locally Euclidean space of dimension $2d$.

Let $\mathscr F$ be a collection of co-ordinate systems $\{(U_\alpha,\phi_\alpha) : \alpha \in A\}$ such that:
 * $\displaystyle \bigcup_{\alpha \in A} U_\alpha = M$

If $\phi_\alpha \circ \phi_\beta^{-1}$ is of biholomorphic as a map $\phi_\beta\left(U_\alpha \cap U_\beta\right) \subseteq \R^{2d} \simeq \C^d \to \phi_\alpha\left(U_\alpha \cap U_\beta\right) \subseteq \R^{2d} \simeq \C^d$ for all $\alpha,\beta \in A$ then we call $\mathscr F$ a complex analytic differentiable structure of dimension $d$ on $M$.