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Bernstein–Sato polynomials of arbitrary varieties

Published online by Cambridge University Press:  04 December 2007

Nero Budur ,
Mircea Mustaţa  and
Morihiko Saito
Nero Budur
Affiliation:
Department of Mathematics, The Johns Hopkins University, Baltimore, MD 21218, USAnbudur@math.jhu.edu
Mircea Mustaţa
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USAmmustata@umich.edu
Morihiko Saito
Affiliation:
RIMS Kyoto University, Kyoto 606-8502, Japanmsaito@kurims.kyoto-u.ac.jp
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Abstract

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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.

Keywords

Bernstein–Sato polynomialV-filtrationmultiplier idealrational singularity32S40
Type
Research Article
Information
Compositio Mathematica , Volume 142 , Issue 3 , May 2006 , pp. 779 - 797
Copyright
Foundation Compositio Mathematica 2006