On complex homogeneous singularities

TL;DR

This paper investigates the singularities of complex homogeneous polynomials at the origin by analyzing monodromy characteristic polynomials, exploring their relation to the Hodge spectrum, and providing detailed insights especially for the case when n=2.

Contribution

It offers new analysis connecting monodromy polynomials with Hodge spectra for complex homogeneous singularities, extending understanding especially for the case n=2.

Findings

Relation between monodromy zeta function and Hodge spectrum established

Detailed study of $ abla_1(t)$ for n=2

Insights into multiplier ideals and local systems in singularity analysis

Abstract

In this article, we consider the singularity of an arbitrary homogeneous polynomial with complex coefficients $f(x_0,\dots,x_n)$ at the origin of $\mathbb C^{n+1}$, via the study of the monodromy characteristic polynomials $\Delta_l(t)$, and the relation between the monodromy zeta function and the Hodge spectrum of the singularity. We go further with $\Delta_1(t)$ in the case $n=2$. This work is based on knowledge of multiplier ideals and local systems.