Which complex manifolds admit a collection of compatible 'charts' $\varphi_{\alpha} : U_{\alpha} \to \mathbb{C}^n$ which cover $X$?

A complex manifold of dimension $n$ is a $2n$-dimensional topological manifold $X$ together with a complex atlas which is a collection of compatible charts $\varphi_{\alpha} : U_{\alpha} \to B(0, 1) \subseteq \mathbb{C}^n$ which cover $X$.

Which complex manifolds $X$ admit a collection of compatible 'charts' $\varphi_{\alpha} : U_{\alpha} \to \mathbb{C}^n$ which cover $X$?

In the case of smooth manifolds, it doesn't matter whether we use the unit ball in $\mathbb{R}^n$ or all of $\mathbb{R}^n$ for the range of the charts because they are diffeomorphic. However, the unit ball in $\mathbb{C}^n$ and $\mathbb{C}^n$ are not biholomorphic, so the distinction is important. Every complex manifold which admits charts with range $\mathbb{C}^n$ also admits charts with range the unit ball (just restrict to the preimage of the ball). However, there are complex manifolds which do not admit charts with image $\mathbb{C}^n$. For example, the unit disc $\mathbb{D} \subset \mathbb{C}$.

Some examples of manifolds which do admit such charts are complex Euclidean spaces $\mathbb{C}^n$, complex projective spaces $\mathbb{CP}^n$ and more generally complex grassmanians $\operatorname{Gr}(k, n)$.

Added Later: I just realised that there is a more natural way of phrasing my question:

Which $n$-dimensional complex manifolds admit neighbourhoods biholomorphic to $\mathbb{C}^n$ for each of their points?

Solution 1:

Your examples about projective spaces and Grassmannian manifolds remind me of a large class of complex manifolds that admit a neighborhood biholomorphic to $\mathbb C^n$ at every point, they are rational homogeneous varieties.

A rational homogeneous variety is formed by $G/P$ where $G$ is a simply connected semi simple complex Lie group and $P$ and a parabolic subgroup. Bruhât decomposition of the semi simple Lie group gives a holomophic cellular decomposition of the homogeneous variety which implies that the homogeneous variety admits an open dense subset biholomorphic to$\mathbb C^n$. Now since the homogeneous variety is transitive under a holomophic group action, every point admits a neighborhood biholomorphic to $\mathbb C^n$. Projective spaces and Grassmannian varieties correspond to the special case when $G$ is of type $A$ and $P$ maximal parabolic subgroups.

Some details can be found in a note by Dennis M.Snow, see theorem 3.1 and context https://www3.nd.edu/~snow/Papers/HomogVB.pdf