In this paper we survey the role of D-module theory in the comparison between logarithmic and meromorphic de Rham complexes of integrable logarithmic connections with respect to free divisors, and we present some new linearity conditions on the Jacobian ideal which arise in this setting.

Linearity conditions on the Jacobian ideal and logarithmic--meromorphic comparison for free divisors

A duality approach to the symmetry of Bernstein–Sato polynomials of free divisors

Given a nonzero germ h of holomorphic function on (C^n,0), we study the condition: ``the ideal Ann\_D 1/h is generated by operators of order 1''. When h defines a generic arrangement of hypersurfaces with an isolated singularity, we show that it is verified if and only if h is weighted homogeneous and -1 is the only integral root of its Bernstein-Sato polynomial. When h is a product, we give a process to test this last condition. Finally, we study some other related conditions.

On meromorphic functions defined by a differential system of order 1, II

Let X be a complex analytic manifold, D ⊂ X a Koszul free divisor with jacobian ideal of linear type (e.g. a locally quasi-homogeneous free divisor), j : U = X −D →֒ X the corresponding open inclusion, E an integrable logarithmic connection with respect to D and L the local system of the horizontal sections of E on U . In this paper we prove that the canonical morphisms ΩX(log D)(E(kD)) −→ Rj∗L, j!L −→ Ω • X(log D)(E(−kD)) are locally isomorphisms in the derived category of sheaves of complex vector spaces for k ≫ 0.

/pdf/on-the-logarithmic-comparison-theorem-for-integrable-3n0cjge2qf.pdf

On the logarithmic comparison theorem for integrable logarithmic connections

Given a nonzero germ h of holomorphic function on (C n ,0), we study the condition: "the ideal AnnD 1/h is generated by operators of order 1" When h defines a generic arrangement of hypersurfaces with an isolated singularity, we show that it is verified if and only if h is weighted homogeneous and −1 is the only integral root of its Bernstein-Sato polynomial When h is a product, we give a process to test this last condition Finally, we study some other related conditions

/pdf/on-meromorphic-functions-defined-by-a-differential-system-of-183hxnz5fv.pdf

On meromorphic functions defined by a differential system of order $1$

Let X be a complex analytic manifold and D \subset X a free divisor. Integrable logarithmic connections along D can be seen as locally free {\cal O}_X-modules endowed with a (left) module structure over the ring of logarithmic differential operators {\cal D}_X(\log D). In this paper we study two related results: the relationship between the duals of any integrable logarithmic connection over the base rings {\cal D}_X and {\cal D}_X(\log D), and a differential criterion for the logarithmic comparison theorem. We also generalize a formula of Esnault-Viehweg in the normal crossing case for the Verdier dual of a logarithmic de Rham complex.

L. Narvaez-Macarro

Papers

Dualit\'e et comparaison sur les complexes de de Rham logarithmiques par rapport aux diviseurs libres