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On the logarithmic comparison theorem for integrable logarithmic connections. (English) Zbl 1166.32005
Authors’ abstract: Let \(X\) be a complex analytic manifold, \(D\subset X\) a free divisor with Jacobian ideal of linear type (for example, a locally quasi-homogeneous free divisor), \(j: U = X - D {\hookrightarrow} X\) the corresponding open inclusion, \(\mathcal E\) an integrable logarithmic connection with respect to \(D\) and \(\mathcal L\) the local system of the horizontal sections of \(\mathcal E\) on \(U\).
In this paper, we prove that the canonical morphisms \(\Omega_X^\bullet (\log D)(\mathcal E(kD))\to Rj_*\mathcal L, j_{!}\mathcal L \to \Omega_X^\bullet(\log D)(\mathcal E(-kD))\) are isomorphisms in the derived category of sheaves of complex vector spaces for \(k \gg 0\) (locally on \(X\)).

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)
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