Bernstein–Sato polynomials of arbitrary varieties

By Nero Budur, M. Mustaţă, Morihiko Saito. Compositio Mathematica, 2004

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We introduce the notion of the 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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/43052ec5df3773f147366a0194063078757c1476

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Citations in the corpus

Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension (2021, background): “We expect that, in general, p(Z) can be described in terms of the Bernstein-Sato polynomial of Z studied in [6] (see Conjecture 3.”
Discreteness and rationality of F-jumping numbers on singular varieties (2009, background): “irational 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 multiplier id”