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

Question

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.

Answer 1

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.