Let $\underline{\Omega}\_X^{\bullet}$ denote the Deligne -- Du Bois complex of a normal variety $X$. What kind sigularities satisfy $gr^k\underline{\Omega}\_X^{\bullet}[k] \simeq \Omega\_X^{[k]}$ where $\Omega\_X^{[k]} := j\_\*\Omega^k\_{X^{\operatorname{reg}}}$ and $j\colon X^{\operatorname{reg}} \hookrightarrow X$ is the inclusion of the regular locus of $X$? ADDED LATER: Is there even a map like $\Omega\_X^{[k]} \to gr^k\underline{\Omega}\_X^{\bullet}[k]$ to begin with?


Recall that the case $k=0$ is by definition the Du Bois singularities and in order to define it, one does not need to assume that $X$ is normal. Log canonical singularities (normal) are known to be Du Bois (<https://arxiv.org/abs/0902.0648>).


EDIT: As Donu pointed out, when $X$ is smooth $h^{-k}\omega\_X^{\bullet} = 0$ for $k\neq -\dim X$ but $gr^k\underline{\Omega}\_X^{\bullet}[k] = \Omega\_X^k$. So the following question does not make sense.
One could perhaps ask more generally when is $gr^k\underline{\Omega}\_X^{\bullet}[k] \simeq h^{-k}\omega\_X^{\bullet}$ where $\omega\_X^{\bullet}$ is the dualizing complex of $X$?



Is there a generalization of Du Bois singularities?
Good question. Your first question has a positive answer when $X$ has finite quotient singularities, i.e. locally analytically $\mathbb{C}^n/G$, with $G\subset GL\_n(\mathbb{C})$ finite. This was essentially proved in Du Bois' original paper "Complexe de de Rham filtré...", see also Steenbrink "Mixed Hodge structure on the vanishing cohomology". I have a hunch (see added comment) this true when $G$ is reductive. There are some other classes as well, like toric singularities. I can say more later, or perhaps Sándor will.


**Added** A better reference for examples would be the book by Navarro Aznar et. al. "Hyperrésolutions cubiques et descente cohomologique".


Also I overlooked your second question: Think about the case when $X$ is smooth. It's almost never true. Perhaps you meant something else?


**2nd edit** I suppose I may as well make my old  [notes](https://www.math.purdue.edu/~arapura/preprints/singform.pdf)  public. See conjecture 2.8 of that for a more precise statement of what I'm expecting.


