Suppose that $X$ is a smooth projective variety $/\mathbb{C}$ with a $\mathbb{C}^{\*}$-action with isolated fixed points. Must $X$ be rational?



Does a torus action with isolated fixed points imply rational?
Yes. This follows from the Białynicki-Birula decomposition (see Theorem 4.4 in [the original paper](https://www.jstor.org/stable/1970915)). 


