The LCT–strong Euler-homogeneity conjecture for free divisors

Let $X$ be a complex analytic manifold of dimension $n$, and let $D$ be a free divisor, meaning that $\operatorname{Der}_X(-\log D)$ is a locally free $\mathcal{O}_X$-module. The Logarithmic Comparison Theorem for $D$ is the assertion that the natural morphism

$$\Omega_X^\bullet(\log D)\longrightarrow Rj_*\mathbb{C}_U$$

is a quasi-isomorphism, where $U=X\setminus D$ and $j:U\hookrightarrow X$ is the inclusion. A divisor is strongly Euler-homogeneous if, at every point $x\in D$, a reduced local equation $f$ admits a derivation $\delta\in\mathfrak{m}_{X,x}\operatorname{Der}_{X,x}$ such that $\delta(f)=f$.

LCT–strong Euler-homogeneity conjecture. If $D$ satisfies the Logarithmic Comparison Theorem, then $D$ is strongly Euler-homogeneous.

This conjecture links the cohomological behavior of the complement of a free divisor with the existence of Euler-type logarithmic derivations. It was proved in dimension $n=2$; the general higher-dimensional case remains open in the source.