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

TL;DR

This paper explores the deep connections between multiplier ideals, the b-function, and the spectrum of hypersurface singularities, providing new formulas and relations that extend existing theorems in singularity theory.

Contribution

It establishes new links between roots of the b-function and jumping coefficients, generalizes formulas for multiplier ideals, and offers alternative proofs for known results in hyperplane arrangements.

Findings

Certain roots of the b-function are jumping coefficients up to a sign.

Explicit formula for multiplier ideals in locally conical divisors.

Alternative proof of Walther's formula on the b-function of hyperplane arrangements.

Abstract

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 Mustata 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.