Exercise with complex projective line

Question

Let $x$, $y$, $z$, $t$ be the standard coordinates for $\mathbb R^4$ and $f:S^3 \to \mathbb {CP}^1$ , with $f(x,y,z,t)=[x+iy:z+it]$. I have to show that $f$ is $C^\infty$ and, given $p:=(1,0,0,0),$ calculate $df_p$ in the coordinates. I have no clue how to do this exercise, because I don't even know what does the notation with the square brackets mean; I didn't find any result searching on Google neither. Can you explain how the function $f$ is defined? Thank you in advance

Answer 1

The brackets are homogeneous coordinates. Recall how projective spaces are defined (will do it in the complex case here): You start with $\mathbb{C}^{n+1} \setminus \lbrace 0 \rbrace$ and then quotient out the group action of $\mathbb{C}^{\times}$ operating by multiplication. Thus elements of $\mathbb{P}^n(\mathbb{C})$ are given by equivalence classes of non-zero $(n+1)$-tuples, where you change representatives by multiplying with non-zero complex numbers. These equivalence classes are the brackets. For example we have $[1:2:3:i] = [i:2i:3i:-1] \in \mathbb{P}^3(\mathbb{C})$. By that we achieved that a point in the projective space corresponds to a line in $\mathbb{C}^{n+1}$, namely the $1$-dimensional subspace generated by any representative of the point.

Do you understand that map now?