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.



Reference on an equivariant resolution of singularities
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.).


