We study the Du Bois complex (Omega) under bar (center dot)(Z) of a hypersurface Z in a smooth complex algebraic variety in terms of its minimal exponent (alpha) over tilde (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 (alpha) over tilde (Z) (Omega) under bar (p)(Z) is an isomorphism, where (Omega) under bar (p)(Z) 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 (alpha) over tilde (Z) > p >= 2, we obtain non-vanishing results for some higher cohomologies of (Omega) under bar (n-p)(Z).
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