The Du Bois complex of a hypersurface and the minimal exponent

Abstract

We study the Du Bois complex ${\underset{\_}{\mathrm{\Omega }}}_Z^{\bullet }$ of a hypersurface Z in a smooth complex algebraic variety in terms of its minimal exponent $\tilde{\mathit{\alpha }}(Z)$. The latter is an invariant of singularities, defined as the negative of the greatest root of the reduced Bernstein–Sato polynomial of Z, and refining the log-canonical threshold. We show that if $\tilde{\mathit{\alpha }}(Z)\ge p+1$, then the canonical morphism ${\mathrm{\Omega }}_Z^p\to {\underset{\_}{\mathrm{\Omega }}}_Z^p$ is an isomorphism, where ${\underset{\_}{\mathrm{\Omega }}}_Z^p$ is the pth associated graded piece of the Du Bois complex with respect to the Hodge filtration. On the other hand, if Z is singular and $\tilde{\mathit{\alpha }}(Z)>p\ge 2$, we obtain non-vanishing results for some higher cohomologies of ${\underset{\_}{\mathrm{\Omega }}}_Z^{n-p}$.