Bernstein–Sato polynomials of arbitrary varieties

Abstract

We introduce the notion of Bernstein-Sato polynomial of an arbitrary variety (which is not necessarily reduced nor irreducible), using the theory of V -filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration by multiplier ideals coincides essentially with the restriction of the V -filtration. This implies a relation between the roots of the Bernstein-Sato polynomial and the jumping coefficients of the multiplier ideals, and also a criterion for rational singularities in terms of the maximal root of the polynomial in the case of a reduced complete intersection. These are generalizations of the hypersurface case. We can calculate the polynomials explicitly in the case of monomial ideals.

Zeta functions and Bernstein–Sato polynomials for ideals in dimension two

“…In the case of an ideal, this follows from the theory of the V -filtration of Kashiwara and Malgrange. Budur, Mustaţǎ, and Saito [6] proved that the definition of b I does not depend on the choice of generators for the ideal I. We refer to [25] for an introduction to the theory of b-functions.…”

“…Not so obvious was the generalization of the concepts of local monodromy and b-functions to several polynomials, this was done by Verdier [29] for local monodromy and by Sabbah [24] and more recently by Budur et al [6] for Bernstein-Sato polynomials. The conjectures we mentioned can still be stated in this broader context.…”

“…In view of the affirmative answer to Question 1.13 in the principal ideal case, we may suppose that I cannot be generated by only one monomial; i.e., we may assume a < c and b > d. Figure 2(a-b) shows the Newton polyhedron of I and the 6 Although this is not necessary for the current argumentation, we may assume this inequality to be strict, because, if mult P (g) were zero, there would be no reason for this blowing-up. where…”

Spectrum and multiplier ideals of arbitrary subvarieties

“…The multiplier ideals were originally defined for any subvariety of a smooth variety (see e.g. [16]), and the b-function for an arbitrary variety has been defined in [4]. In this paper we introduce the spectrum for an arbitrary variety generalizing Steenbrink's definition [26], [27].…”

“…, f r are local generators of the ideal of X, see [16]. They are closely related to the filtration V of Kashiwara [14] and Malgrange [18], see [4]. Set…”