Question

As is very well known, the algebraic variety $S^2$ is isomorphic to projective variety $\mathbb{CP}^1$ as a complex manifold. As is also well known, the coordinate ring of $S^2$ is given by $< x,y,z > / < x^2 + y^2 +z^2 - 1 >$ and the function field of $\mathbb{CP}^1$ is $\mathbb{C}(\mathbb{CP}^1)$ (its coordinate ring being $\mathbb{C}$). My question is how the coordinate ring of $S^2$ and the function field of $\mathbb{CP}^1$ are related? Presumably this relation is a special case of a general variety-complex manifold relationship.

Answer 1

Any projective variety is also a real affine variety, by using the real and imaginary parts of the coordinates $x_{jk} = z_j\overline{z_k}$. You should first normalize the projective coordinates to have Hermitian-Euclidean length 1. Ordinarily the projective coordinates are sections of a line bundle that is only defined up to a scalar. But with the length normalization, the only ambiguity is the phase. The phase cancels out in the definition of $x_{jk}$, so it is a well-defined function rather than just a section.

This realification of $\mathbb{CP}^n$ is a map to $(n+1) \times (n+1)$ Hermitian matrices. It is important in quantum probability: The image of the map is the set of pure states of a quantum system; its convex hull is the set of all states. This convex region is the quantum analogue of the simplex of states (= measures = distributions) for classical probability on a finite set. The matrices have unit trace, so the image lies in an $(n^2+2n)$-dimensional real subspace. You can check that it is a sphere when $n=1$.

Another viewpoint that is important is that of toric varieties. The phase part of the toric action on $\mathbb{CP}^1$ is rotation about the $z$ axis, and the moment map is the projection onto the $z$ coordinate. This too generalizes to any projective toric variety.