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nLab > nLab General Discussions: Complex manifold

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    • CommentRowNumber1.
    • CommentAuthorColin Tan
    • CommentTimeJun 21st 2014

    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?

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 21st 2014

    Personally I’d call it complex nn-space. Ultimately we want that the derivatives of transition functions for an atlas induce cocycle data U i∩U j→GL(ℂ n)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 nn-dimensional vector space” if complex nn-space doesn’t seem precise enough.

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeJun 21st 2014

    You could say ‘complexified cartesian space’.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 21st 2014

    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 nC^n,

    and then

    One must use the open unit disk in C nC^n as the model space instead of C nC^n because these are not isomorphic, unlike for real manifolds.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 21st 2014

    That’s a good point, David.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2014
    • (edited Jun 21st 2014)

    Yes, disks. We have been talking about that extensively in various other entries recently. I have fixed it in the entry here.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 21st 2014

    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.

    • CommentRowNumber8.
    • CommentAuthorZhen Lin
    • CommentTimeJun 21st 2014

    As Wikipedia says, Liouville’s theorem implies that every holomorphic map from complex space ℂ n\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.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 21st 2014

    So what should the definition be on the page complex manifold?

    • CommentRowNumber10.
    • CommentAuthorZhen Lin
    • CommentTimeJun 21st 2014

    I don’t know. My lecturer used the definition where general open subsets of ℂ n\mathbb{C}^n are used as local models. However, one notes that every open subset of ℂ n\mathbb{C}^n can be covered by either nn-dimensional open balls (resp. polydiscs) of different sizes, and that nn-dimensional open balls (resp. polydiscs) of different sizes are isomorphic, so it would seem that either one can be used as a local model.

    • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 21st 2014

    Zhen Lin is right: either balls or polydisks can be used as local models, and one gets the same notion of manifold either way.

    • CommentRowNumber12.
    • CommentAuthorColin Tan
    • CommentTimeJul 17th 2014

    I took up Toby’s suggestion went ahead to call ℂ n{\mathbb{C}}^n “complex n-dimensional complexified carteisan space” instead of the original “complex n-dimensional complex line”.

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 17th 2014

    I don’t much like the way it reads (too heavy), but it’s an improvement over what was there before.

    • CommentRowNumber14.
    • CommentAuthorColin Tan
    • CommentTimeJul 17th 2014
    Would complexified (n-dimensional Cartesian space) be more readable?
    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 17th 2014

    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 nn-space” was quite enough, but if one really feels a strong urge to utter “cartesian space”, then I think one could say “complexified nn-dimensional cartesian space” without that second pair of parentheses. Still sounds heavy to me, but I could learn to live with it.

    • CommentRowNumber16.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 18th 2014

    I went ahead and made the change (along with some other tiny edits made earlier today).

    • CommentRowNumber17.
    • CommentAuthorTobyBartels
    • CommentTimeJul 18th 2014

    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 nn-dimensional cartesian space’ (‘complexified nn-dimensional cartesian space’ with links) should be sufficient. (So might ‘nn-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 nn-space’ or even ‘ℂ n\mathbb{C}^n’ will usually be clear enough.

    • CommentRowNumber18.
    • CommentAuthorTobyBartels
    • CommentTimeJul 18th 2014

    Seeing this terminology in context at real space, I recall that another way to say this would be ‘complex nn-dimensional affine space’ or ‘nn-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.

    • CommentRowNumber19.
    • CommentAuthorColin Tan
    • CommentTimeJul 19th 2014

    How about ’complexified cartesian n-space’? If we choose this, then eventually we can lighten the reference to ℝ n{\mathbb{R}}^n from ’n-dimensional cartesian space’ to ’cartesian n-space’.

    • CommentRowNumber20.
    • CommentAuthorTobyBartels
    • CommentTimeJul 20th 2014

    You could do that, but it's harder to link. The first reference can be lengthy; every subsequent reference is simply ‘ℂ n\mathbb{C}^n’.

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