# $\mathbb{C}^*$-equivariant smooth completion of a quasiprojective variety

A famous theorem by Sumihiro states that, given a normal quasi-projective variety $$X$$ with a regular $$G$$-action (where $$G$$ is a connected linear algebraic group), there is a G-equivariant projective embedding $$X\hookrightarrow \mathbb{P}^n,$$ where $$G$$ acts on $$\mathbb{P}^n$$ linearly. The closure of the image of this map thus gives a $$G$$-equivariant completion of $$X,$$ i.e. a projective variety with a $$G$$-action which has a $$G$$-invariant open subvariety isomorphic to $$X.$$

Now, in the case that the quasiprojective variety $$X$$ over $$\mathbb{C}$$ is also nonsingular, and the group $$G$$ is equal to $$\mathbb{C}^*,$$ is it true that one can always find a nonsingular $$\mathbb{C}^*$$-equivariant completion of $$X$$? I have seen some authors using this result without explanation/references, so I am wondering where it follows from.

Since you seem to work in characteristic zero (even over $$\mathbb{C}$$), you can first take any $$\mathbb{C}^\times$$-equivariant completion, and then take its canonical resolution of singularities (see, e.g., Bierstone, E. and P. Milman, “Functoriality in resolution of singularities”, Research Institute for Mathematical Sciences Publications, 44, no. 2 (2008): 609–39); it is automatically $$\mathbb{C}^\times$$-equivariant.
• Great! As it is not stated in that paper explicitly, I suppose that the $\mathbb{C}^*$-equivariance follows from the functionality property, that is, the morphism action $\mathbb{C}^* \times X \rightarrow X$ can be lifted to a morphism $\widetilde{\mathbb{C}^* \times X} = \mathbb{C}^* \times \tilde{X} \rightarrow \tilde {X}$? Here $X$ is the completion and $\tilde{X}$ is its canonical resolution. Commented Oct 19, 2021 at 22:18