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.

  • $\begingroup$ Thank you for the comment. I added the definition. They are the tangent sheaves. $\endgroup$
    – tarosano
    Commented Jul 12, 2011 at 7:42

2 Answers 2


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.).


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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