The Du Bois complex of a hypersurface and the minimal exponent

Abstract

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 $\tilde{\alpha }(Z)$. 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 $\tilde{\alpha }(Z)\ge p+1$, 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 $\tilde{\alpha }(Z)>p\ge 2$, we obtain non-vanishing results for some of the higher cohomologies of ${\underline{\Omega }}_Z^{n-p}$.

arXiv comment18 pages, v.2 : Lemma 2.1 was added and an argument in the proof of Theorem 1.1 was fixed, v.3: improved exposition, final version, to appear in Duke Math. J