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Cohomology of the complement of a free divisor. (Cohomologie du complémentaire d’un diviseur libre.) (French) Zbl 0844.32020

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)].

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)

Citations:

Zbl 0731.32005
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