Complex atlas - Imported Knowlpedia Content

Definition

Let $M$ be a Hausdorff, second-countable topological manifold of real dimension $2n$ . A complex atlas of complex dimension $n$ on $M$ is a family of complex coordinate charts $\{(U_\alpha,\varphi_\alpha)\}$ whose domains cover $M$ and such that, whenever $U_\alpha\cap U_\beta\neq\varnothing$ , the transition map

\varphi_\beta\circ\varphi_\alpha^{-1}

is holomorphic on $\varphi_\alpha(U_\alpha\cap U_\beta)$ . Two complex atlases are equivalent when their union is again a complex atlas. A complex-manifold structure is equivalently an equivalence class of complex atlases, or the unique maximal atlas containing any representative.

Maximal atlases and equivalence

Every complex atlas is contained in a unique maximal one: add every complex coordinate chart compatible with all charts already present. Two atlases generate the same maximal atlas exactly when their union is compatible. Thus changing to a smaller covering atlas does not change the resulting complex manifold Huybrechts, §2.1.

Examples and non-examples

The single identity chart on an open subset of $\mathbb C^n$ is a complex atlas. The standard affine charts on complex projective space form another. By contrast, a covering by complex coordinate charts is not a complex atlas if even one overlap map is merely smooth but not holomorphic.

Conventions and scope

Some authors use “complex atlas” only for a maximal atlas; others, as here, allow any compatible covering family and say “maximal complex atlas” when maximality matters. The equivalence-class formulation removes this terminological difference Forster, §1.

References

D. Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Springer DOI record. Relevant: §2.1, complex manifolds and holomorphic atlases.
O. Forster, Lectures on Riemann Surfaces, Springer, 1981. Springer DOI record. Relevant: §1, complex charts and atlases.