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:

$(1): \quad \ds \bigcup_{\alpha \mathop \in A} U_\alpha = M$

$(2): \quad$ For all $\alpha, \beta \in A$, $\phi_\alpha \circ \phi_\beta^{-1}$ is, as a mapping ${\phi_\beta} \sqbrk {U_\alpha \cap U_\beta} \to {\phi_\alpha} \sqbrk {U_\alpha \cap U_\beta}$, biholomorphic.

$(3): \quad$ 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 $\struct {U, \phi} \in \mathscr F$.

Also see

• Definition:Complex Manifold

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