On Euler-homogeneity for free divisors

TL;DR

This work addresses whether a free divisor $D$ with Logarithmic Comparison Theorem ($LCT$) must be strongly Euler-homogeneous. It develops an intrinsic Jordan decomposition of singular derivations into commuting semisimple and nilpotent parts and proves that this decomposition preserves the logarithmic structure, enabling new proofs of the $n=2$ and $n=3$ cases. A key result is that if $D$ satisfies $LCT$ and a reduced local equation $f$ is not a product, then some logarithmic derivation must have nonzero trace, linking $ ext{D}$-module criteria to Euler-homogeneity. Using this framework for plane curves and leveraging Koszul-freeness considerations, the authors derive that plane curves satisfying $LCT$ are strongly Euler-homogeneous and obtain an alternative route to the $n=3$ case, suggesting a path toward higher dimensions. Overall, the paper connects logarithmic, Euler-homogeneity, and $ ext{D}$-module structures to advance understanding of when $LCT$ forces strong Euler-homogeneity in free divisors.

Abstract

In 2002, it was conjectured that a free divisor satisfying the so-called Logarithmic Comparison Theorem must be strongly Euler-homogeneous and it was proved for the two-dimensional case. Later, in 2006, it was shown that the conjecture is also true in dimension three, but, today, the answer for the general case remains unknown. In this paper, we use the decomposition of a singular derivation as the sum of a semisimple and a topologically nilpotent derivation that commute in order to deal with this problem. By showing that this decomposition preserves the property of being logarithmic, we are able to give alternative proofs for the low-dimensional known cases.

Theorems & Definitions (58)

• Definition 1.1
• Definition 1.2
• Conjecture 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Definition 2.1
• Lemma 2.1
• Remark