MULTIPLIER IDEALS, b-FUNCTION, AND SPECTRUM OF A HYPERSURFACE SINGULARITY

We prove that certain roots of the Bernstein-Sato polynomial (i.e. b-function) are jumping coefficients up to a sign, showing a partial converse of a theorem of L. Ein, R. Lazarsfeld, K.E. Smith, and D. Varolin. We also prove that certain roots are determined by a filtration on the Milnor cohomology, generalizing a theorem of B. Malgrange in the isolated singularity case. This implies a certain relation with the spectrum which is determined by the Hodge filtration, because the above filtration is related to the pole order filtration. For multiplier ideals we prove an explicit formula in the case of locally conical divisors along a stratification, generalizing a formula of Mustaţǎ in the case of hyperplane arrangements. We also give another proof of a formula of U. Walther on the bfunction of a generic hyperplane arrangement, including the multiplicity of −1.