Castro-Jiménez, Francisco J.; Mond, David; Narváez-Macarro, Luis Cohomology of the complement of a free divisor. (Cohomologie du complémentaire d’un diviseur libre.) (French) Zbl 0844.32020 C. R. Acad. Sci., Paris, Sér. I 320, No. 1, 55-58 (1995). Let \(X\) be a complex analytic manifold, \(D \subset X\) be a reduced hypersurface, \(U = X \backslash D\), and \(j : U \to X\) be the inclusion. Assume \(D\) is strongly quasihomogeneous, that is, for every point \(p \in D\) there exists a coordinate system centered at \(p\) such that \(D\) is defined by a quasihomogeneous polynomial. Assume also that the \({\mathcal O}_X\)-module \(\Omega^1_X (\log D)\) of meromorphic differential 1-forms with logarithmic poles along \(D\) is free. The authors prove that the natural morphism of complexes \(\Omega^\bullet_X (\log D) \to Rj_* C_U\) is a quasi-isomorphism. Thus, the cohomology of the complement of \(D\) can be described by means of the cohomology of the logarithmic de Rham complex \(\Omega^\bullet_X (\log D)\). It should be noted that for any reduced hypersurface \(D\) the latter complex is quasi-isomorphic to the complex \(\omega^\bullet_D\) of regular meromorphic differential forms with shift of grading by \(- 1\) [see the reviewer, Adv. Sov. Math. 1, 211-246 (1990; Zbl 0731.32005)]. Reviewer: A.G.Aleksandrov (Moskva) Cited in 1 Review MSC: 32S25 Complex surface and hypersurface singularities 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14F40 de Rham cohomology and algebraic geometry 32C35 Analytic sheaves and cohomology groups 14C20 Divisors, linear systems, invertible sheaves 32J05 Compactification of analytic spaces 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) Keywords:analytic manifolds; quasihomogeneous hypersurfaces; normal crossing; free divisors; stable morphisms; discriminants; logarithmic de Rham complex; regular meromorphic differential forms Citations:Zbl 0731.32005 PDFBibTeX XMLCite \textit{F. J. Castro-Jiménez} et al., C. R. Acad. Sci., Paris, Sér. I 320, No. 1, 55--58 (1995; Zbl 0844.32020)