Testing the Logarithmic Comparison Theorem for Free Divisors

Abstract

We propose in this work a computational criterion to test if a free divisor {\small $D\subset {\bf C}^n$} verifies the Logarithmic Comparison Theorem (LCT); that is, whether the complex of logarithmic differential forms computes the cohomology of the complement of {\small $D$} in {\small ${\bf C}^n$}.

For Spencer free divisors {\small $D\equiv(f=0)$}, we solve a conjecture about the generators of the annihilating ideal of {\small $1/f$} and make a conjecture on the nature of Euler homogeneous free divisors which verify LCT. In addition, we provide examples of free divisors defined by weighted homogeneous polynomials that are not locally quasi-homogeneous.