Abstract. Let D, x be a plane curve germ. We prove that the complex Ω • (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 ⊂ C 3 which is not locally weighted homogeneous, but for which this (second) assertion continues to hold.
Let X be a complex analytic manifold, D ⊂ X a free divisor with jacobian ideal of linear type (e.g. a locally quasi-homogeneous free divisor), j : U = X − D ֒→ X the corresponding open inclusion, E an integrable logarithmic connection with respect to D and L the local system of the horizontal sections of E on U. In this paper we prove that the canonical morphisms Ω • X (log D)(E(kD)) − → Rj * L, j ! L − → Ω • X (log D)(E(−kD)) are isomorphisms in the derived category of sheaves of complex vector spaces for k ≫ 0 (locally on X).
Abstract. We find explicit free resolutions for the D-modules Df s and D½s f s =D½s f sþ1 , where f is a reduced equation of a locally quasi-homogeneous free divisor. These results are based on the fact that every locally quasi-homogeneous free divisor is Koszul free, which is also proved in this paper.
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