[Math] Equivariant resolution of singularities

Question

I am looking for some references on equivariant resolution of singularities. In most references quoted on mathoverflow (for instance : Reference on an equivariant resolution of singularities), they only talk about finite group action (if I am not mistaken).

I was wondering if it is known that equivariant resolutions do not exist in general for larger group (are there counter-examples?). I am especially interested when the group is $\mathbb{C}^*$.

Thanks in advance.

Answer 1

To any variety $X$ over a field of characteristic zero, say $k$, one can attach a resolution of singularities, say $X'\to X$, with the following properties:

1. $X'\to X$ is an isomorphism over $X_{\mathrm{reg}}$.
2. if $\Theta:X\rightarrow Y$ is an isomorphism of schemes (not necessarily defined over $k$), then $\Theta$ can be extended to an isomorphism $\Theta':X'\rightarrow Y'$ compatible with $\Theta$ and the resolutions.

It follows from 1. and 2. that if $\Psi:Y\rightarrow Z$ is a second isomorphism of schemes, then $(\Psi\circ\Phi)'$ and $\Psi'\circ\Phi'$ coincide on the inverse image of $X_{\mathrm{reg}}$ in $X'$, and hence they coincide on $X'$. With a similar argument, we have $(\mathrm{id}_X)'=\mathrm{id}_{X'}$. In particular, if $G$ is any group of automorphisms of $X$ (not necessarily defined over $k$), the action of $G$ on $X$ can be lifted to an action on $X'$.

For a reference, see:

O. Villamayor: Equimultiplicity, algebraic elimination, and blowing-up. Adv. in Math. 262 (2014) 313-369. (ArXiv link, DOI link)