The Du Bois complex of a hypersurface and the minimal exponent

Abstract

We study the Du Bois complex Ω_Z∙ of a hypersurface Z in a smooth complex algebraic variety in terms of its minimal exponent α˜(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 α˜(Z)≥p+1, then the canonical morphism ΩZp→Ω_Zp is an isomorphism, where Ω_Zp 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 α˜(Z)>p≥2, we obtain non-vanishing results for some higher cohomologies of Ω_Zn−p.