The map from a complex torus to the projective algebraic curve

  • #26
mathwonk
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This is explained in the post #23 of zinq. I.e.a closed ball in C^2 contains a topological copy of C^2, i.e. the interior of the ball, but with a boundary which is a 3 sphere, whereas P^2 is a copy of C^2 plus a "boundary" P^1 which is a 2 sphere. Thus there is a continuous map from the closed ball to P^2, mapping the 3 sphere boundary of the ball, to the 2 sphere "boundary", or rather the 2 sphere "at infinity", of P^2, by the Hopf map S^3-->S^2. This is not a bijective map, since it drops dimension. The fibers of the Hopf map are circles. This is called "attaching" a 4 cell to the 2 sphere. In general (complex) P^n, a real 2n dimensional manifold, only has cells in even (real) dimensions up to 2n, i.e. it has cells in real dimensions 2,4,....,2n, which are attached by the construction zinq described, and summarized in his last sentence. If you have seen the concept of topological homology, complex P^n has zero homology groups in odd dimensions, and one dimensional homology groups in (non negative) even dimensions up to 2n. Nonetheless, the continuous surjection from the compact closed ball in C^2, onto P^2 does prove P^2 is also compact.

Indeed we cannot have P^2 homeomorphic to a closed ball, since then it would not be a manifold, but would be a "manifold with boundary". The P^1 which is added at infinity to C^2 to make P^2, is not a boundary in the sense of manifold theory, since it has the wrong dimension. The representation of P^2 just given, as the image of a closed ball does not make it obvious that P^2 is actually a manifold, i.e. everywhere locally Euclidean, but that can be seen from the more symmetric definition of P^2 as complex lines through the origin of C^3. I.e. the P^1 at infinity can be taken to be the lines contained in any choice of fixed plane through the origin.

E.g. with coordinates x,y,z on C^3, the line at infinity can be chosen as lines in either the x,y plane or the y,z plane, or the z,x plane. The 3 complements of those 3 choices, gives a cover of P^2 by three different copies of C^2. E.g. the set of complex lines through the origin but not lying in the x,y (i.e. z=0) plane, is bijectively equivalent to the set of points of the plane z=1, which is a copy of C^2. I.e. each complex line through the origin but not lying in z=0, contains exactly one point whose z coordinate is 1.
 
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  • #27
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Here's how I like to think of ℂℙ2: Pretend you start out at any point p ∈ ℂℙ2 and consider what happens when you consider the 4-dimensional ball Br of radius r > 0 in ℂℙ2 centered at p as r increases. Its boundary ∂Br = S3 is a 3-dimensional sphere.

Geometrically, any point of ℂℙ2 looks like any other point of it.

(Now, ℂℙ2 is the quotient of S5 = {(u,v,w) ∈ ℂ3 | |u|2 + |v|2 + |w|2 = 1} by the equivalence relation

(u,v,w) ~ (αu, αv, αw)​

for any α ∈ ℂ with |α| = 1. So a typical point p ∈ ℂℙ2 is represented by the set {(α, 0, 0) ∈ ℂ3 | |α| = 1} ⊂ S5 . The points of ∈ ℂℙ2 that are farthest from the point p are represented by the set {(0, v, w) ∈ ℂ3 | |v|2 + |w|2 = 1} ⊂ S5, which is recognizable as a copy of S3.)

Back to the expanding 4-dimensional ball Br: When this ball, centered at p represented by {(α, 0, 0)}, reaches its maximum radius within ℂℙ2, its boundary ∂Br (represented in S5) will be that exact copy of S3, namely {(0, v, w) ∈ ℂ3 | |v|2 + |w|2 = 1}. But in ℂℙ2 this is not a 3-dimensional sphere, because instead it is a union of equivalence classes. Instead, in ℂℙ2 this is the quotient of S3 by the equivalence relation that we have been looking at — which makes the S3 into a ℂℙ1. Or in familiar terms, ℂℙ1 is a 2-dimensional sphere S2.

Summary: ℂℙ2 is a 4-dimensional ball whose boundary S3 has been quotiented by the eqivalence relation (v,w) ~ (αv, αw) for any α ∈ ℂ with |α| = 1 — which turns the boundary of the 4-ball into a copy of S2.

It's not intuitively obvious, but it's true, that this quotient of a closed 4-dimensional ball is indeed a (4-dimensional) manifold.
 
  • #28
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In #27 above, I meant to write B(r) for the closed ball of radius r in CP2, but somehow ended up putting the r as a superscript (resulting in Br instead).
 

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