The Du Bois complex of a hypersurface and the minimal exponent

TL;DR

This paper investigates the relationship between the Du Bois complex of a hypersurface and its minimal exponent, revealing conditions under which the complex simplifies and establishing non-vanishing results for certain cohomology groups.

Contribution

It establishes a precise link between the minimal exponent and the isomorphism of the Du Bois complex's graded pieces, advancing understanding of singularity invariants.

Findings

Isomorphism of $oldsymbol{ ext{Omega}_Z^p}$ and $oldsymbol{ ext{underline{ extOmega}_Z^p}}$ when minimal exponent $oldsymbol{ ilde{ extalpha}(Z)}$ is at least $oldsymbol{p+1}$.

Non-vanishing of higher cohomologies of $oldsymbol{ extunderline{ extOmega}_Z^{n-p}}$ for singular hypersurfaces with $oldsymbol{ ilde{ extalpha}(Z)>p ext{ and }p extgreater 1}$.

Refinement of the relationship between singularity invariants and the structure of the Du Bois complex.

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 $\widetilde{\alpha}(Z)$. 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 $\underline{\Omega}_Z^{n-p}$.