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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.

Abstract. In an earlier version of this paper written by the second named author, we showed that the jumping coefficients of a hyperplane arrangement depend only on the combinatorial data of the arrangement as conjectured by Mustaţǎ. For this we proved a similar assertion on the spectrum. After this first proof was written, the first named author found a more conceptual proof using the Hirzebruch-Riemann-Roch theorem where the assertion on the jumping numbers was proved without reducing to that for the spectrum. In this paper we improve these methods and show that the jumping numbers and the spectrum are calculable in low dimensions without using a computer. In the reduced case we show that these depend only on fewer combinatorial data, and give completely explicit combinatorial formulas for the jumping coefficients and (part of) the spectrum in the case the ambient dimension is 3 or 4. We also give an analogue of Mustaţǎ's formula for the spectrum.

Abstract. For an effective divisor on a smooth algebraic variety or a complex manifold, we show that the associated multiplier ideals coincide essentially with the filtration induced by the filtration V constructed by B. Malgrange and M. Kashiwara. This implies another proof of a theorem of L. Ein, R. Lazarsfeld, K.E. Smith and D. Varolin that any jumping coefficient in the interval (0,1] is a root of the Bernstein-Sato polynomial up to sign. We also give a refinement (using mixed Hodge modules) of the formula for the coefficients of the spectrum for exponents not greater than one or greater than the dimension of the variety minus one. IntroductionLet X be a smooth complex algebraic variety or a complex manifold, and D be an effective divisor on X with a defining equation f . The multiplier ideal J (D) is a coherent sheaf of ideals of the structure sheaf O X , and can be defined by using an embedded resolution. This is defined also for the Q-divisors αD with α > 0, and we get a decreasing family {J (αD)} α∈Q , where J (αD) = O X for α ≤ 0. By construction there exist positive rational numbers 0 < α 1 < α 2 < · · · such that J (α j D) = J (αD) = J (α j+1 D) for α j ≤ α < α j+1 where α 0 = 0, see (1.1). These numbers α j (j > 0) are called the jumping coefficients (or numbers) of the multiplier ideals associated to D. We define the graded pieces G(D, α) to be J ((α − ε)D)/J (αD) for 0 < ε ≪ 1.Let i f : X → Y := X × C denote the embedding by the graph of f , and t be the coordinate of C. Malgrange [20] constructed the filtration V on B f and M. Kashiwara [15] did it in a more general case, see also [16]. We index V decreasingly by rational numbers so that the action of ∂ t t − α on GrThis is actually an immediate consequence of [24], 3.5 and [23], 3.3.17. Indeed, the normal crossing case was proved in the former (see also (2.3) below) and the general case

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