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.