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At complex manifold, a complex manifold is defined as a manifold modeled on C^n (the complex n -dimensional complex line).
In my understanding, the complex line refers to C^1. I am not sure what to replace this by. Should we call C^n “complex Cartesian space” or “complex affine space”? Is there a name by which C^n already goes by in the Lab?
Personally I’d call it complex $n$-space. Ultimately we want that the derivatives of transition functions for an atlas induce cocycle data $U_i \cap U_j \to GL(\mathbb{C}^n)$, so it’s the automorphisms on the complex vector space that enter here; therefore you could say “complex $n$-dimensional vector space” if complex $n$-space doesn’t seem precise enough.
You could say ‘complexified cartesian space’.
The wikipedia entry makes a point of saying that
a complex manifold is a manifold with an atlas of charts to the open unit disk in $C^n$,
and then
One must use the open unit disk in $C^n$ as the model space instead of $C^n$ because these are not isomorphic, unlike for real manifolds.
That’s a good point, David.
Yes, disks. We have been talking about that extensively in various other entries recently. I have fixed it in the entry here.
Hmm. Why does wikipedia say
The following spaces are different as complex manifolds
and list complex space; the unit disk or open ball; the polydisk?
It says the open ball is the right model.
As Wikipedia says, Liouville’s theorem implies that every holomorphic map from complex space $\mathbb{C}^n$ to either the open ball or the polydisc must be constant. So complex space is not isomorphic to either the open ball or the polydisc. The open ball is not isomorphic to the polydisc either – see here.
So what should the definition be on the page complex manifold?
I don’t know. My lecturer used the definition where general open subsets of $\mathbb{C}^n$ are used as local models. However, one notes that every open subset of $\mathbb{C}^n$ can be covered by either $n$-dimensional open balls (resp. polydiscs) of different sizes, and that $n$-dimensional open balls (resp. polydiscs) of different sizes are isomorphic, so it would seem that either one can be used as a local model.
Zhen Lin is right: either balls or polydisks can be used as local models, and one gets the same notion of manifold either way.
I took up Toby’s suggestion went ahead to call ${\mathbb{C}}^n$ “complex n-dimensional complexified carteisan space” instead of the original “complex n-dimensional complex line”.
I don’t much like the way it reads (too heavy), but it’s an improvement over what was there before.
Of course this is mostly a matter of aesthetics. Personally I think one could omit the entire parenthetical without any loss of comprehension whatever, and I also think prose ought to sound close to what one might say aloud. I therefore thought “complex $n$-space” was quite enough, but if one really feels a strong urge to utter “cartesian space”, then I think one could say “complexified $n$-dimensional cartesian space” without that second pair of parentheses. Still sounds heavy to me, but I could learn to live with it.
I went ahead and made the change (along with some other tiny edits made earlier today).
I agree with Todd #15. It can be helpful to have the term ‘cartesian space’ sitting there to help one link to cartesian space. In that case, ‘complexified $n$-dimensional cartesian space’ (‘complexified $n$-dimensional cartesian space’ with links) should be sufficient. (So might ‘$n$-dimensional complexified cartesian space’, but this could be ambiguous about how we are counting dimensions.) If we don't need to spell everything out, then ‘complex $n$-space’ or even ‘$\mathbb{C}^n$’ will usually be clear enough.
Seeing this terminology in context at real space, I recall that another way to say this would be ‘complex $n$-dimensional affine space’ or ‘$n$-dimensional affine space over the complex numbers’. This is less specific, since complexified cartesian space is only one of the many complex affine spaces (fixing a dimension); this is because a cartesian space comes equipped with coordinates. If you're not worrying about (even iso‑) morphisms, however, then there's no difference, since every affine space may be so equipped.
How about ’complexified cartesian n-space’? If we choose this, then eventually we can lighten the reference to ${\mathbb{R}}^n$ from ’n-dimensional cartesian space’ to ’cartesian n-space’.
You could do that, but it's harder to link. The first reference can be lengthy; every subsequent reference is simply ‘$\mathbb{C}^n$’.
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