Logarithmic cohomology of the complement of a plane curve

Abstract

Let D, x be a plane curve germ. We prove that the complex \( \Omega^\bullet(\log D)_x \) computes the cohomology of the complement of D, x only if D is quasihomogeneous. This is a partial converse to a theorem of [5], which asserts that this complex does compute the cohomology of the complement, whenever D is a locally weighted homogeneous free divisor (and so in particular when D is a quasihomogeneous plane curve germ). We also give an example of a free divisor \( D\subset \mathbb {C}^3 \) which is not locally weighted homogeneous, but for which this (second) assertion continues to hold.