On 1-forms on isolated complete intersection curve singularities

TL;DR

This paper reviews classical results on holomorphic 1-forms on complex curve singularities and investigates the relationship between Milnor and Tjurina numbers in plane curve singularities.

Contribution

It compiles classical results on holomorphic 1-forms and explores the inequality between Milnor and Tjurina numbers for isolated plane curve singularities.

Findings

Collected classical results on holomorphic 1-forms on curve singularities

Analyzed the pull-back of 1-forms under normalization

Proposed the inequality 3μ < 4τ for plane curve singularities

Abstract

We collect some classical results about holomorphic 1-forms of a reduced complex curve singularity. They are used to study the pull-back of holomorphic 1-forms on an isolated complete intersection curve singularity under the normalization morphism. We wonder whether the Milnor number $\mu$ and the Tjurina number $\tau$ of any isolated plane curve singularity satisfy the inequality $3\mu <4\tau$.