Spectrum and multiplier ideals of arbitrary subvarieties

Dimca, Alexandru; Maisonobe, Philippe; Saito, Morihiko

Annales de l'Institut Fourier, Tome 61 (2011) no. 4, pp. 1633-1653.

Nous introduisons un spectre pour des sous-variétés arbitraires.Ceci généralise la définition de Steenbrink pour les hypersurfaces. Dans le cas d’une singularité isolée d’intersection complète, il coïncide au spectre donné par Ebeling et Steenbrink, sauf pour les coefficients des exposants entiers. Nous montrons une relation avec les gradués des idéaux multiplicateurs en utilisant la filtration V de Kashiwara et Malgrange. Ceci implique une généralisation partielle d’un théorème de Budur dans le cas des hypersurfaces. Le point clef est de considérer la somme directe des gradués d’un idéal multiplicatif comme un module sur l’algèbre définissant le cône normal de la sous-variété. Nous donnons aussi une description combinatoire dans le cas des idéaux monomiaux.

We introduce a spectrum for arbitrary subvarieties. 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 the filtration V of Kashiwara and Malgrange. This implies a partial generalization of a theorem of Budur in the hypersurface case. The key 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.

Classification : 32S40, 32S35
Keywords: Spectrum, V-filtration, multiplier ideal
Mot clés : spectre, V-filtration, idéal multiplicateur

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