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Most elementary book use terminology complex plane for the same notion (integrating in complex plane, for instance). This is like usual double standard: Riemann surface is just a complex curve.
The complex plane refers to $\mathbb{C}$ itself, regarded as a 2d $\mathbb{R}$-vector space, whereas complex line is anything equivalent over $\mathbb{C}$ with $\mathbb{C}$ as a $\mathbb{C}$-vector space. That’s why we speak of complex line bundles as opposed to complex plane bundles.
So the complex plane but a complex line?
And the generalized the doesn't apply; as a complex line, the complex plane has nontrivial automorphisms.
So the complex plane but a complex line?
Yes. Take any reference on complex lines, for instance the first paragraph here.
the generalized the doesn’t apply
True, agreed.
The complex plane refers to $\mathbb{C}$ itself, regarded as a 2d $\mathbb{R}$-vector space
You might want that way, but on the contrary to this principle so many (in fact most) standard one variable complex analysis books in chapters on residues and their applications call it complex plane and use holomorphic functions there (and no use of vector space structure) all the time, hence really mean it inhabited with complex numbers rather than real vectors. Also, as you know with most physics references and the books on Green functions and related PDE and ODE methods.
I agree about complex line bundles. In that context, the terminology as Urs put it is absolutely standard (this is advanced subject and the difference must hence be in mind of geometric users, what is not in usual basic analysis and physics),
If you regard $\mathbb{C}$ as a complex vector space then it is 1-dimensional hence a kind of “complex curve”, not a plane.
But in any case, the point here is that a complex line is a 1-d $\mathbb{C}$-vector space without being equipped with an identification with $\mathbb{C}$ itself. So “complex plane” is a different notion from “complex line”.
If you regard $\mathbb{C}$ as a complex vector space then it is 1-dimensional hence a kind of “complex curve”, not a plane.
Urs, I’m pretty sure Zoran gets that. His point about established terminology remains valid, whether or not you think it’s logical terminology.
Plus, I’m not sure why you insist that “complex plane” means being equipped with a specific identification with $\mathbb{C}$. It seems to me that much of the time such terminology refers to objects in the holomorphic category (sample: the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere) where that theorem doesn’t specify for example an origin for “the complex plane” – nor for that matter are we here considering the complex plane qua 2-d real vector space (cf. comment 3).
In fact, it seems to me that “complex plane” need not refer to a specific category at all – the same piece of terminology people use to refer to an isomorphism type either in the holomorphic category, or the real vector space category (?) as you seem to say, or whatever, to circumvent having to think of a new piece of terminology for each such situation. That might not be nice from an nPOV, but it’s a sociological reality nonetheless. Disagree?
Disagree?
Hm, yes. If we didn’t know where $i$ is in the complex plane, we couldn’t speak for instance of the “upper half plane” inside it.
(The statement of the uniformization theorem seems unrelated to this, as it is about the existence of arbitrary equivalences.)
But maybe I was brought up with different literature than you. Which book talks about making choices of points 0, 1 and $i$ in the “complex plane”? And what does such a book say for the “complex plane” equipped with these choices?
(Edited because I wish to rethink my response)
Looking back at #2: I think that in particular in contour integration, everyone uses explicitly the structure of the complex plane as being $\mathbb{C}$. This is necessary to even make sense of standard expressions like $\oint \frac{1}{z} d z$. There is no contour integration technique in an arbitrary complex line. The letter “$z$”, being a coordinate, is the hallmark of an explicit identification with $\mathbb{C}$.
Urs, surely we each see the point the other is making. Of course I agree with you that things like “upper half plane” and “contour integrals” have coherent meaning only if we have a specific ambient $\mathbb{C}$ in mind. This might be seen a specific case of the general principle that it isn’t really coherent to speak of properties of subobjects unless one has a fixed containing object $S$ in mind.
The point I was trying to make is that (I think) “complex plane” is also a catch-all phrase people use that can refer also to an isomorphism type (analogous to how people speak of “the” field with 9 elements), or even to different categories (holomorphic, quasi-conformal, etc.) where a choice of basis $\{1, i\}$ is irrelevant. I thought people spoke of “the complex plane” to cover all of these varying contexts.
I don’t know, I don’t see this. So you are also saying that the entry complex line should be changed?
I don’t think that’s right. And I find you and Zoran might want to provide some kind of predecessor for this in the literature, for I am not aware of any.
But I won’t further fight about this.
I don’t know, I don’t see this. (…) I don’t think that’s right. And I find you and Zoran might want to provide some kind of predecessor for this
Sorry, I’m not sure which claim(s) you are objecting to. What do “this” and “that” refer to specifically?
I don’t really feel like “fighting” either, and I suspect in the end we don’t disagree as much as you think we are. It’s not a deep point I’m making; it’s analogous to the point that “real line” can refer to lots of different contexts, and particularly lots of different categories, depending on the context. (As for the entry complex line, I’d have to think about what changes, if any, I’d want.)
It’s late over there! Get some sleep! :-)
Note also that ’complex line’ can, in complex algebraic geometry, refer to an embedded $\mathbb{P}^1$.
What do “this” and “that” refer to specifically?
We were discussing the claim in #2 that the notions “complex line” and “complex plane” are the same.
I see; thanks. I agree with you (and Zoran, second paragraph of #7) that complex line, as in complex line bundle, is effectively as you describe. And I see how you are pushing back against #2 in your comment #12, and I do agree with you that there’s a useful distinction to be made here.
My objection (and I think Zoran’s too, in the first paragraph of #7, but he will correct me if I got him wrong) is that, again I think, people use the phrases “complex line” and “complex plane” also to cover other situations and other categories. Sometimes vector spaces, but also sometimes algebro-geometrically ($Spec \mathbb{C}[x]$; maybe here people usually say “line over $\mathbb{C}$”), or sometimes holomorphically, sometimes quasi-conformally, etc. I just didn’t want an nLab entry to lock us into saying that “complex line/plane” has to mean just one of these categorical contexts. The point however is somewhat orthogonal to the useful distinction you were trying to make.
17: Urs, I did not make a claim that these NOTIONS are the same. Just the distinction in the terminology the way you put it is not consistently used in the classical references. Hence, depending on the context and community, people may use plane even precisely for the case where you use line. This is not fighting, but the discussion of the actual dynamics of the terminology, where people often lean toward naive intuition and ignore building everything into perfect overall systematics. I think we should record a variety of patterns in usage of terminology, to guide ourselves into it. It is good to know that terminology differs in certain patterns across the subfields and subcommunities. Nobody really uses entirely formal language without simplifications and abuses of notation and language to make it more palatable.
Much of the literature, as you often protest does the definitions and theorems in math in formal way without explaining their origin. The Idea or Motivation section of the nLab, when present, tries to rectify this pattern within our realm. Similarly we could guide ourselves in the terminological universe.
I like your discussion in 12 about contour integration.
I feel like ‘the complex plane’ and ‘the real line’ refer to specific objects $\mathbb{C}$ and $\mathbb{R}$, but they refer also to various objects in categories $E$ by abuse of terminology (and one that I gladly engage in). More specifically, we have a trivial category whose object is denoted ‘$\mathbb{C}$’, and this is equipped with various functors to other categories. Since by rights a functor is just an anafunctor, the functor to $E$ really defines a clique in $E$, not a specific object. But I tend to think of all of these objects, not only within one category but across all categories, as one thing.
ETA: I'm not sure whether this means that I agree with Todd or with Urs!
I hope we can define the complex line in synthetic differential geometry. Let R be the real line, which is a ring object. My attempt is that the complex line C is a R-module object which has 2 dimensions. I wish C to be R-algebra object. For this, equip C with an R-module endomorphism J such that $J^2 = - \mathop{id}_C$, the complex strcture, which allows me to define multiplication.
If (C,J) is a complex line, then so is (C, -J). To disambiguate the holomorphicity of a smooth morphism $f:C\to C$, I will need to choose complex structures on both the source and target.
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