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

By Morihiko Saito. Compositio Mathematica, 2004

54 citations recorded by the source corpus.

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 $-1$.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/94273d020c1a908115cc9c5e20f2000993cb1c29

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Discreteness and rationality of F-jumping numbers on singular varieties (2009, background): “tudy of birational geometry; see for example [Kol97]. More generally, the sequence of jumping numbers contains a great deal of information about the singularities of Xand elements of a, see [ELSV04], [Sai07], [BMS06], [Tuc08]. In characteristic p>0, the test ideal τ(X,at) is an analog of the mult”
Jumping coefficients and spectrum of a hyperplane arrangement (2009, background): “Using [6], we can show for any holomorphic function f that the converse also holds if α is an isolated jumping coefficient (i.e. if α is not a jumping coefficient for D \ {0}), see Proposition (4.2) below.”