Spectrum and multiplier ideals of arbitrary subvarieties

TL;DR

This paper introduces a new spectrum concept for arbitrary varieties, extending previous definitions for hypersurfaces and linking it to multiplier ideals and algebraic structures, with special cases for monomial ideals.

Contribution

It generalizes the spectrum notion to all varieties, relates it to multiplier ideals, and provides a combinatorial description for monomial ideals.

Findings

Spectrum for arbitrary varieties is introduced.

Relation established between spectrum and multiplier ideals.

Provides a combinatorial description for monomial ideals.

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.