Multiplier ideals, $b$-function, and spectrum of a hypersurface singularity

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. 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 Mustaţ̌a in the case of hyperplane arrangements. We also give another proof of a formula of U. Walther on the $b$-function of a generic hyperplane arrangement, including the multiplicity of $-1$.