Definition:Complex Analytic Differentiable Structure

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Definition

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

Then a complex analytic differentiable structure $\mathscr F$ on $M$ is a collection of charts $\set {\struct {U_\alpha, \phi_\alpha} : \alpha \in A}$ such that:

$(1): \quad \ds \bigcup_{\alpha \mathop \in A} U_\alpha = M$
$(2): \quad \phi_\alpha \circ \phi_\beta^{-1}$ is of biholomorphic as a map $\map {\phi_\beta} {U_\alpha \cap U_\beta} \to \map {\phi_\alpha} {U_\alpha \cap U_\beta}$ for all $\alpha, \beta \in A$



$(3): \quad$ If $\struct {U, \phi}$ is a chart such that $\phi \circ \phi_\alpha^{-1}$ and $\phi_\alpha \circ \phi^{-1}$ are biholomorphic for all $\alpha \in A$, then $\struct {U, \phi} \in \mathscr F$.


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