# zbMATH — the first resource for mathematics

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)
Full Text: