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Duality and comparison for logarithmic de Rham complexes with respect to free divisors. (Dualité et comparaison pour les complexes de de Rham logarithmiques par rapport aux diviseurs libres.) (French) Zbl 1089.32003
Let $$X$$ be a complex analytic manifold of dimension $$n$$, $$D \subset X$$ a free divisor (i.e., such that the sheaf $$\text{Der} (\log D)$$ of vector fields logarithmic with respect to $$D$$ is $$\mathcal{O}_X$$-free, $$\mathcal{O}_X(*D)$$ the sheaf of meromorphic functions with poles along $$D$$, $$\mathcal{D}(\log D)$$ the 0-term of a $$V$$-filtration of Malgrange-Kashiwara with respect to $$D$$ of the sheaf $$\mathcal{D}_X$$ of differential operators. $$\mathcal{D}_X(\log D)$$ is a coherent sheaf with noetherian fiber of finite cohomological dimension. An integrable logarithmic conexion (along $$D$$) is a left $$\mathcal{D}_X(\log D)$$-module which is an locally free $$\mathcal{O}_X$$-module of finite rank. Finally, such a connexion $$\mathcal{E}$$ is called admissible if the complex $$\mathcal{D}_X\bigotimes^L_{\mathcal{D}_X(\log D)} \mathcal{E}$$ id concentrates in 0 degree and is a holonomic $$\mathcal{D}_X$$-module. When $$D$$ is a free Koszul divisor any integrable logarithmic connexion is admissible. If we denote by $$\omega_X$$ (respectively $$\omega_x(\log D)$$) the sheaf of holomorphic $$n$$-forms (respectively with logarithmic poles along $$D$$) then the main result of the authors is the following: For any integrable logarithmic $$\mathcal{E}$$ there is a natural isomorphism in the derived category of right $$\mathcal{D}_X$$-modules $R \text{Hom}_{\mathcal{D}_X}(\mathcal{D}_X \mathop{\otimes}\limits^{L} {}_{ \mathcal{D}_X(\log D)} \mathcal{E}, \mathcal{D}_X) \simeq \omega_X \otimes_{0_X} (\mathcal{D}_X \mathop{\otimes}\limits^{L} {}_{\mathcal{D}_X(\log D)} \mathcal{E}^* (D))[-n],$ $$(\mathcal{E}^* = \text{Hom}_{0_X}(\mathcal{E}, 0_X)$$ is the dual connexion), when $$D$$ is a free divisor. As an application, from this theorem the authors describe the Verdier dual of the logarithmic de Rham complex of an integrable logarithmic connexion when $$D$$ is Koszul, which is the generalization of the same result in case $$D$$ with normal crossing due to H. Esnault and E. Viehweg [Invent. Math. 86, 161–194 (1986; Zbl 0603.32006)]. Different interesting corollaries are obtained; for instance the logarithmic de Rham complex is perverse iff the divisor $$D$$ is such that $$\mathcal{O}_X$$ is admissible. Another generalization of a result de Esnault-Viehweg [loc. cit.] is the following. Suppose $$\mathcal{E}$$ is a logarithmic integrable connexion (with respect to $$D$$) such that its $$\mathcal{E}^*$$ is admissible. Then there is a natural isomorphism in the derived category $\Omega_X^\bullet (\log D)(\mathcal{E}) \simeq \Omega_X^\bullet (\log D) (\mathcal{E}^* (-D))^V$ where $$\Omega_X^\bullet (\log D)$$ is the logarithmic de Rham complex and $$''V''$$ denotes the Verdier dual. The authors obtain also a differential characterization and proof of the logarithmic comparison theorem CTCL [F. J. Calderón Moreno, D. Mond, L. Narváez Macario and J. Castro Jiménez, Commun. Math. Helv. 77, No. 1, 24–38 (2002; Zbl 1010.32016)]. One says that TCL is tame for the divisor $$D$$ if the inclusion $$\Omega_X^\bullet (\log D) \hookrightarrow \Omega_X^\bullet(*D)$$ is an isomorphism. They prove that TCL is tame for $$D$$ iff the morphism $$\rho : \mathcal{D}_X \bigotimes^L _{\mathcal{D}_X(\log D)} \mathcal{O}_X(D) \to \mathcal{O}_X(*D)$$ is an isomorphism in the derived category where $$\rho$$ is defined as $$\rho (P \otimes a) P(a)$$. (In particular for the TCL to hold true for $$D$$ it is necessary that integrable logarithmic connexion $$\mathcal{O}_X(D)$$ be admissible, or equivalently that $$\mathcal{O}_X$$ be admissible.) If $$D$$ is a free divisor locally-quasihomogeneous then the inclusion $$\Omega_X^\bullet (\log D) \hookrightarrow \Omega_X^\bullet (*D)$$ is an quasi-isomorphism. Some interesting examples are worked out.
Some interesting questions are still open; for example if $$D \subset X$$ is a free divisor, the logarithmic de Rham complex is analytically constructible? (problem 5.4); or if $$\mathcal{O}_X$$ is admissible it results that any logarithmic integrable conexion is admissible (problem 5.5).

##### MSC:
 32C38 Sheaves of differential operators and their modules, $$D$$-modules 14F40 de Rham cohomology and algebraic geometry 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 32S20 Global theory of complex singularities; cohomological properties
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