# Equivariant resolution of singularities

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 $\mathbb{C}^*$.

• Equivariant resolutions exist for the action of any algebraic group. See, for example, 3.9.1 in Kollar's book "Lectures on resolution of singularities".
– naf
Commented Sep 26, 2013 at 10:58
• @ulrich Thanks! I still wonder why many papers only deal with finite group actions. But that does not really matter, now I have this reference. Commented Sep 26, 2013 at 12:10
• I've need to reference this in a paper and used: Cor 7.6.3, O. E. Villamayor U. Patching local uniformizations. Ann. Sci. E ́cole Norm. Sup. (4), 25(6):629–677, 1992. Commented Sep 26, 2013 at 18:00
• @ulrich In the proof of the proposition 3.9.1, it seems that for Kollar, a group action is, in particular a smooth morphism $G\times X\rightarrow X$, or am I mistaken ? (or maybe for him smooth morphism only means that any fiber is smooth??)
– pi_1
Commented Aug 18, 2020 at 12:28

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 $$X'$$.