Hairy ball theorem : a counter example ?

Question

Recall the hairy ball theorem : any continuous vector field on a sphere $S^n$ of even dimension must vanish at least once.

Now consider the Riemann sphere $\mathbb{C} \cup \infty \simeq S^2$. Let $X(z)=(1+|z|^2) \frac{d}{dz}$. Then $X$ is a continuous, complex non-vanishing vector field of the Riemann sphere (unless I'm mistaken). You can check it doesn't vanish at infinity by using the coordinates $w=1/z$ for $w$ near $0$.

I guess that this does not contradict the hairy ball theorem because here we view $S^2$ as a complex manifold of complex dimension one instead of a real 2-manifold.

In particular, in the chart $z$, the expressions of the continuous vector fields on the Riemann sphere are exactly the $X = h(z) \frac{d}{dz}$ with $h$ a continuous complex-valued function which is $O(z^2)$ at infinity (there are no additional restriction like "it has to vanish at least once").

Am I being correct here ?

Answer 1

Be careful here. You're using a smooth section (but not a holomorphic one) of $T\Bbb CP^1\otimes\Bbb C$. But it is not continuous at $\infty$. When you change coordinates, as you suggested, using $w=1/z$, you get $X=-\big(w^2+\dfrac w{\overline w}\big)\dfrac{\partial}{\partial w}$, which, in fact, has no limiting value at $w=0$.

Answer 2

Although, as @Ted pointed out, your example isn't continuous at $\infty$, it's interesting to note that there is a smooth nonvanishing complex vector field on the Riemann sphere. Here's an example: $$ X = \left( 1-z^2\right) \frac{\partial}{\partial z} + \left( 1+\overline z^2\right)\frac{\partial}{\partial \overline z}. $$ This vanishes nowhere in $\mathbb C$. To see what happens at $\infty$, make the change of coordinates $z=1/w$. It ends up looking almost the same: $$ X = \left( 1-w^2\right) \frac{\partial}{\partial w} -\left( 1+\overline w^2\right)\frac{\partial}{\partial \overline w}, $$ which is smooth and nonvanishing at $w=0$.

What's going on here? Basically, a complex vector field can be thought of as a pair of real vector fields (the real and imaginary parts). For a complex vector field to vanish, both its real and imaginary parts have to vanish. So I just chose a smooth vector field on the sphere whose real part vanishes only at $\infty$ and whose imaginary part vanishes only at $0$.