Higher Du Bois singularities of hypersurfaces

For a complex algebraic variety $X$, we introduce higher $p$-Du Bois singularity by imposing canonical isomorphisms between the sheaves of K\"ahler differential forms $\Omega_X^q$ and the shifted graded pieces of the Du Bois complex $\underline{\Omega}_X^q$ for $q\le p$. If $X$ is a reduced hypersurface, we show that higher $p$-Du~Bois singularity coincides with higher $p$-log canonical singularity, generalizing a well-known theorem for $p=0$. The assertion that $p$-log canonicity implies $p$-Du Bois has been proved by Mustata, Olano, Popa, and Witaszek quite recently as a corollary of two theorems asserting that the sheaves of reflexive differential forms $\Omega_X^{[q]}$ ($q\le p$) coincide with $\Omega_X^q$ and $\underline{\Omega}_X^q$ respectively, and these are shown by calculating the depth of the latter two sheaves. We construct explicit isomorphisms between $\Omega_X^q$ and $\underline{\Omega}_X^q$ applying the acyclicity of a Koszul complex in a certain range. We also improve some non-vanishing assertion shown by them using mixed Hodge modules and the Tjurina subspectrum in the isolated singularity case. This is useful for instance to estimate the lower bound of the maximal root of the reduced Bernstein-Sato polynomial in the case where a quotient singularity is a hypersurface and its singular locus has codimension at most 4.