wikipedia definition of complex manifolds

Question

On wikipedia it states that the charts of a complex manifold should have image in the open unit disk in $\mathbb{C}^n$. see: https://en.wikipedia.org/wiki/Complex_manifold

I disagree with this, shouldn't the image be in an open set of $\mathbb{C}^n$ like Wolfram says?... is Wikipedia wrong? consider the projective space as a complex manifold and its cover by $D(x_i)= \{[x_0...x_n] \mid x_i \neq 0\}$

for n=1 Riemann mapping theorem doesn't work for unbounded domains.

Answer 1

I don't know why the Wikipedia article insists on charts taking their images in the unit disk. This is certainly not necessary.

Just as with smooth structures, there are different ways of defining a complex structure, all of which are equivalent. In all cases, a complex structure is determined by an atlas of charts whose transition maps are biholomorphisms. We can require that the image of each chart is

1. the open unit ball in $\mathbb C^n$,
2. some open subset of the open unit ball in $\mathbb C^n$, or
3. some open subset of $\mathbb C^n$.

Clearly any atlas of type 1 is also an atlas of type 2, and any atlas of type 2 is also an atlas of type 3. If we are given an atlas of type 3, then we can construct an atlas of type 1 simply by replacing each chart $(U,\phi)$ with the collection $\{(U_\alpha,\phi_\alpha)\}$, where $U_\alpha$ is any set of the form $\phi^{-1}(B_\alpha)$ for a ball $B_\alpha\subset\mathbb C^n$, and $\phi_\alpha$ is $\phi|_{U_\alpha}$ followed by an affine map taking $B_\alpha$ onto the unit ball.

The one thing we cannot do is insist that all the charts have images equal to $\mathbb C^n$. For example if $\mathbb D\subseteq\mathbb C$ is the unit disk and $(U,\phi)$ is a holomorphic chart for $\mathbb D$ whose image is all of $\mathbb C$, then $\phi^{-1}\colon \mathbb C\to U$ is a bounded entire function and thus constant.