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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.

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    $\begingroup$ 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". $\endgroup$
    – naf
    Commented Sep 26, 2013 at 10:58
  • $\begingroup$ @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. $\endgroup$
    – Libli
    Commented Sep 26, 2013 at 12:10
  • $\begingroup$ 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. $\endgroup$
    – Jim Bryan
    Commented Sep 26, 2013 at 18:00
  • $\begingroup$ @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??) $\endgroup$
    – pi_1
    Commented Aug 18, 2020 at 12:28

1 Answer 1

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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)

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