- #1
camel_jockey
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I am trying to understand all details of the complex projective space CPn. Since surely CP1 must be the most simply to understand, I started out with it and even there I cannot gain full understanding. I would be eminently thankful for any help :D
Nearly all texts trying to describe CPn get very general about it, using mostly language and not so much explicitness. Perhaps they assume some foreknowledge that I as a physicist lack :)
I was wondering if someone could shed some light on this.
1) Firstly, neither on Wikipedia nor on Wolfram Mathworld can I find a clear, thorough and detailed definition of what a complex line is. Is it a collection of points of dimension 2? How does one parametrize it in the same way that a real line (x,y) in the plane has a parametrization (x(t),y(t)) for some interval of t.
2) Furthermore, since complex lines are fundamentally different to real lines, I have no idea what a complex line "through the origin" is supposed to mean.
3) Do we identify ANY two points in C^(n+1) that differ by ANY multiplicative factor lambda? Because from what I understand, any complex point z can be moved to any other complex point w if a suitable lambda is found, just use
lambda = z / w
This would mean that all points in C^(n+1) are identified.
I hope that when I spell it out this way, you guys can see where I have gone wrong.
4) To specify a "complex line" in CPn, what is the minimum amount of information you will need to uniquely determine 1 exact "line" ? (points on the manifold need to be unique and fully defined).
Nearly all texts trying to describe CPn get very general about it, using mostly language and not so much explicitness. Perhaps they assume some foreknowledge that I as a physicist lack :)
I was wondering if someone could shed some light on this.
1) Firstly, neither on Wikipedia nor on Wolfram Mathworld can I find a clear, thorough and detailed definition of what a complex line is. Is it a collection of points of dimension 2? How does one parametrize it in the same way that a real line (x,y) in the plane has a parametrization (x(t),y(t)) for some interval of t.
2) Furthermore, since complex lines are fundamentally different to real lines, I have no idea what a complex line "through the origin" is supposed to mean.
3) Do we identify ANY two points in C^(n+1) that differ by ANY multiplicative factor lambda? Because from what I understand, any complex point z can be moved to any other complex point w if a suitable lambda is found, just use
lambda = z / w
This would mean that all points in C^(n+1) are identified.
I hope that when I spell it out this way, you guys can see where I have gone wrong.
4) To specify a "complex line" in CPn, what is the minimum amount of information you will need to uniquely determine 1 exact "line" ? (points on the manifold need to be unique and fully defined).