The Du Bois complex of a hypersurface and the minimal exponent

We study the Du Bois complex $\underline{\Omega}_Z^\bullet$ of a hypersurface $Z$ in a smooth complex algebraic variety in terms its minimal exponent $\widetilde{\alpha}(Z)$. 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 $\widetilde{\alpha}(Z)\geq p+1$, then the canonical morphism $\Omega_Z^p\to \underline{\Omega}_Z^p$ is an isomorphism, where $\underline{\Omega}_Z^p$ is the $p$-th associated graded piece of the Du Bois complex with respect to the Hodge filtration. On the other hand, if $Z$ is singular and $\widetilde{\alpha}(Z)>p\geq 2$, we obtain non-vanishing results for some of the higher cohomologies of $\underline{\Omega}_Z^{n-p}$.