Complex manifold

Holomorphic Maps

In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc^[1] in the complex coordinate space $\mathbb {C} ^{n}$ , such that the transition maps are holomorphic.

The term complex manifold is variously used to mean a complex manifold in the sense above (which can be specified as an integrable complex manifold), and an almost complex manifold.

^ One must use the open unit disc in the $\mathbb {C} ^{n}$ as the model space instead of $\mathbb {C} ^{n}$ because these are not isomorphic, unlike for real manifolds.

[1] One must use the open unit disc in the $\mathbb {C} ^{n}$ as the model space instead of $\mathbb {C} ^{n}$ because these are not isomorphic, unlike for real manifolds.

[1]