Spectrum and multiplier ideals of arbitrary subvarieties

Abstract

We introduce a spectrum for arbitrary varieties. This generalizes the definition by Steenbrink for hypersurfaces. In the isolated complete intersection singularity case, it coincides with the one given by Ebeling and Steenbrink except for the coefficients of integral exponents. We show a relation to the graded pieces of the multiplier ideals by using a relation to the filtration V of Kashiwara and Malgrange. This implies a partial generalization of a theorem of Budur in the hypersurface case. The point is to consider the direct sum of the graded pieces of the multiplier ideals as a module over the algebra defining the normal cone of the subvariety. We also give a combinatorial description in the case of monomial ideals.

Singularity invariants related to Milnor fibers: survey

“…The Hodge spectrum has a generalization to any subscheme Z in a nonsingular variety X, using Verdier's specialization functor and M. Saito's mixed Hodge modules. There is a relation between the multiplier ideals and the specialization functor, [40].…”

“…The Newton polytope of a monomial ideal determines explicitly the Hodge spectrum [40], the multiplier ideals and the jumping numbers, [81]; the test ideals and F -jumping numbers, [68]; and the p-adic zeta function, [71].…”

On $b$-function, spectrum and multiplier ideals

“…7.5. Corollary (DMS [14]). If n Λ,α = 0 with α ∈ (0, 1), then there is a nonnegative integer j 0 such that α + j ∈ JN(Z) for any j ≥ j 0 ∈ N. 7.6.…”

“…If n Λ,α = 0 with α ∈ (0, 1), then there is a nonnegative integer j 0 such that α + j ∈ JN(Z) for any j ≥ j 0 ∈ N. 7.6. Theorem (DMS [14]). If T is a transversal slice to a stratum of a good Whitney stratification and r = codim T , we have…”

“…8.5. Theorem (Spectrum) (Dimca, Maisonobe, S. [14]). We have a one-to-one correspondence between the maximal compact faces σ of P a and the irreducible components Λ of the fiber (N Z X) 0 , and Sp(Z, Λ) = cσ i=1 e σ t i/cσ .…”