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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.

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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.

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  • $\begingroup$ 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. $\endgroup$
    – Filip
    Commented Oct 19, 2021 at 22:18

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