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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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