Higher Du Bois singularities of hypersurfaces

For a complex algebraic variety (Formula presented.), we introduce higher (Formula presented.) -Du Bois singularity by imposing canonical isomorphisms between the sheaves of Kähler differential forms (Formula presented.) and the shifted graded pieces of the Du Bois complex (Formula presented.) for (Formula presented.). If (Formula presented.) is a reduced hypersurface, we show that higher (Formula presented.) -Du Bois singularity coincides with higher (Formula presented.) -log canonical singularity, generalizing a well-known theorem for (Formula presented.). The assertion that (Formula presented.) -log canonicity implies (Formula presented.) -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 (Formula presented.) ((Formula presented.)) coincide with (Formula presented.) and (Formula presented.), respectively, and these are shown by calculating the depth of the latter two sheaves. We construct explicit isomorphisms between (Formula presented.) and (Formula presented.) 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.