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Duality of Gauß-Manin systems associated to linear free divisors. (English) Zbl 1272.32026
This article is a continuation of the author’s work on certain Gauß-Manin connections associated with linear free divisors, as defined in [I. de Gregorio, D. Mond and C. Sevenheck, Compos. Math. 145, No. 5, 1305–1350 (2000; Zbl 1238.32022)]. Let $$h$$ be a reduced, homogeneous polynomial on $$V := \mathbb{C}^n$$ and $$D = V(h)$$ the associated divisor. When the sheaf of logarithmic vector fields $$\Theta(- \log D)$$ has a free $$\mathcal{O}_V$$-basis given by homogeneous linear vector fields, $$D$$ is said to be a linear free divisor. Moreover, it is said to be reductive if $$G_D := \{ g \in \mathrm{GL}(V) \;| \;g(D) \subset D \}$$ is a reductive group. To a reductive linear free divisor one can associate the top cohomology group $G( * ) = \mathbb{H}^{n-1}\big(\Omega_{V/T}^{\bullet}(* D)[\theta,\theta^{-1}], \theta d - df \wedge \big),$ where $$h : V \rightarrow T = \mathrm{Spec\,} \mathbb C[t]$$, $$f$$ is a generic linear form on $$\mathbb{C}^n$$ and $$\big(\Omega_{V/T}^{\bullet}(* D)[\theta,\theta^{-1}], \theta d - df \wedge\big )$$ is a certain complex of relative differential forms. The finite rank, free $$\mathbb{C}[\theta^{\pm 1}, t^{\pm 1}]$$-module $$G( * D)$$ is equipped with an integrable connection, the “Gauß-Manin connection”, thus making it a $$\mathcal{D}$$-module, where $$\mathcal{D} = \mathbb{C}[\theta^{\pm 1}, t^{\pm 1}] \langle \partial_t, \partial_{\theta} \rangle$$. This $$\mathcal{D}$$-module was defined and extensively studied in [loc. cit.].
The main result of the article under review is the identification $\mathrm{Hom}_{\mathbb{C}[\theta^{\pm 1}, t^{\pm 1}]} \left( G(*D),\mathbb{C}[\theta^{\pm 1}, t^{\pm 1}] \right) \simeq \iota^* G(*D)$ of $$\mathcal{D}$$-modules, where $$\iota$$ is a certain involution on $$\mathrm{Spec} \;\mathbb{C}[\theta^{\pm 1}, t^{\pm 1}]$$. This implies that there exists a non-degenerate pairing $$G(*D) \otimes \iota^* G(* D) \rightarrow \mathbb{C}[\theta^{\pm 1}, t^{\pm 1}]$$. The key to proving this result is an explicit presentation of the $$\mathcal{D}$$-module $$G(*D)$$. This allows the author to construct a free resolution of $$G(*D)$$ and hence to calculate the Verdier dual of $$G(*D)$$.
It is shown that the duality theorem allows one to sharpen several of the main results from [loc. cit.]. In particular, it gives a proof of Conjecture 5.5 from [loc. cit.].

##### MSC:
 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
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##### References:
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