Hodge ideals and spectrum of isolated hypersurface singularities

TL;DR

This paper introduces the Hodge ideal spectrum for isolated hypersurface singularities, compares it with the Steenbrink spectrum, and explores conditions for their coincidence or difference, including examples and special cases.

Contribution

It defines the Hodge ideal spectrum, compares it with the Steenbrink spectrum, and establishes conditions for their equivalence or divergence in various singularity cases.

Findings

Hodge ideal spectrum and Steenbrink spectrum coincide in weighted homogeneous cases.

Sufficient conditions are provided for the spectra to coincide or differ in non-weighted-homogeneous cases.

An example shows Hodge ideals can be non-weakly decreasing modulo the Jacobian ideal.

Abstract

We introduce Hodge ideal spectrum for isolated hypersurface singularities to see the difference between the Hodge ideals and the microlocal $V$-filtration modulo the Jacobian ideal. Via the Tjurina subspectrum, we can compare the Hodge ideal spectrum with the Steenbrink spectrum which can be defined by the microlocal $V$-filtration. As a consequence of a formula of Mustata and Popa, these two spectra coincide in the weighted homogeneous case. We prove sufficient conditions for their coincidence and non-coincidence in some non-weighted-homogeneous cases where the defining function is semi-weighted-homogeneous or with non-degenerate Newton boundary in most cases. We also show that the convenience condition can be avoided in a formula of Zhang for the non-degenerate case, and present an example where the Hodge ideals are not weakly decreasing even modulo the Jacobian ideal.