Higher Du Bois singularities of hypersurfaces

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