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Let $X$ be an algebraic variety over $\mathbb{C}$ (or a normal complex space). I found the word "equivariant resolution" in several papers on singularity theory or deformation theory. I think that it means the birational proper morphism of complex spaces $f: Y \rightarrow X $ where $Y$ is a complex manifold such that $f_{\ast} \Theta_Y \simeq \Theta_X$ where $\Theta_X$ is the tangent sheaf, i.e. the dual of the Kahler differential sheaf on $X$ and $\Theta_Y$ is the tangent sheaf on $Y$.

Question 1 Does that equivariant resolution always exist for an complex algebraic variety $X$?

Question 2 Can $Y$ be taken as an smooth algebraic variety?

If you know the reference, please let me know about it. In Wahl's paper on equisingular deformations, the preprint by Hironaka was cited but I couldn't find it.

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What do you denote by $\Theta_X$ and $\Theta_Y$? –  Sándor Kovács Jul 12 '11 at 0:38
    
Thank you for the comment. I added the definition. They are the tangent sheaves. –  tarosano Jul 12 '11 at 7:42
    
OK, I thought that is what you meant. However, I don't think this is very likely to hold for any birational map. –  Sándor Kovács Jul 12 '11 at 8:05

2 Answers 2

up vote 4 down vote accepted

Two other references:

Bierstone, Edward; Milman, Pierre D. (1997), "Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant.", Invent. Math. 128 (2): 207–302, doi:10.1007/s002220050141, MR1440306

Encinas, S.; Villamayor, O. (1998), "Good points and constructive resolution of singularities.", Acta Math. 181 (1): 109–158, doi:10.1007/BF02392749, MR1654779

From what I understand, both references provide a canonical resolution in characteristic zero which is functorial for smooth morphisms (in particular equivariant, compatible with restriction to open subsets, etc.).

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The word "equivariant" usually refers to doing whatever indicated by also respecting a group action. In the context of your question this means that there is a group action on $X$ and it is lifted to $Y$ such that $f$ is equivariant with respect to the lifted action.

A relevant reference is

MR1453072 (98c:14011) Abramovich, Dan(1-BOST); Wang, Jianhua(1-MIT) Equivariant resolution of singularities in characteristic 0. (English summary) Math. Res. Lett. 4 (1997), no. 2-3, 427–433. 14E15

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