Multiplier ideals, 𝑉-filtration, and spectrum

Abstract

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

$V$-filtrations in positive characteristic and test modules

“…Finally, one obtains an equivalence of the bounded derived category of lfgu Fmodules with the bounded derived category of constructible F p -sheaves by first passing to the corresponding étale site via the pull back along the natural map Xé t → X Zar of sites and then applying the derived Hom functor in the category of lfgu F -modules (see [15,Sections 9,11] for details). Moreover, under this correspondence the abelian category of lfgu F -modules is mapped to perverse constructible So we have reduced to the situation that N ⊆ R n , where we have a commutative diagram…”

Hodge Ideals for -Divisors, -Filtration, and Minimal Exponent

“…with the multiplier ideal I (1 − ǫ)D associated to the Q-divisor (1 − ǫ)D with 0 < ǫ ≪ 1, which measures the failure of the pair (X, D) to be log canonical. On the other hand, when D is integral Budur and Saito [BS05] have shown the identification When D is a reduced divisor (corresponding in the notation above to β = 0 and D = H = Z), Saito showed in [Sai16] that a relationship of this type continues to hold in a weaker sense even for p ≥ 1, namely…”

Poles of the complex zeta function of a plane curve

“…The log-canonical threshold appears as the smallest jumping number. The opposites in sign to the jumping numbers in (0, 1] are always roots of the Bernstein-Sato polynomial [30], see also [21]. The spectral numbers [66,67] are a set of logarithms of the eigenvalues of the monodromy constructed using the mixed Hodge structure of the cohomology of the Milnor fiber.…”