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Monodromy at infinity and Fourier transform. (English) Zbl 0920.14003
From the introduction: Let \(f:U\to\mathbb{C}\) be a regular function on a smooth quasi-projective variety \(U\). For \(t\in\mathbb{C}\), the cohomology spaces \(H^*(f^{-1}(t),\mathbb{Q})\) underly a natural mixed Hodge structure. J. Steenbrink and S. Zucker [Invent. Math. 80, 489-542 (1985; Zbl 0626.14007)] have constructed a limit mixed Hodge structure when \(t\to\infty\). This mixed Hodge structure can also be obtained [cf. M. Saito, Publ. Res. Inst. Math. Sci. 26, No. 2, 221-333 (1990; Zbl 0727.14004)] by compactifying \(f\) as a map: \(F:{\mathcal X}\to\mathbb{P}^1\) with \({\mathcal X}\) smooth, and by constructing a mixed Hodge module structure on the nearby cycles at \(t= \infty\) of the sheaf \(R\kappa_*\mathbb{Q}_U\), if \(\kappa:U\hookrightarrow{\mathcal X}\) denotes the inclusion: One obtains the Steenbrink-Zucker limit by taking the global de Rham complex of this mixed Hodge module on \(F^{-1}(\infty)\).
This paper proposes to recover this limit mixed Hodge structure using Fourier transform techniques. The main object is the \({\mathcal D}_{\mathcal X}[\tau] \langle \partial_t \rangle\)-module \(\kappa_+{\mathcal E}^{-\tau f}\), where \({\mathcal E}^{-\tau f}\) is \({\mathcal O}_U[\tau]\) equipped with the natural \({\mathcal D}_{\mathcal X}[\tau] \langle \partial_\tau \rangle\)-action twisted by \(e^{-\tau f}\), and \(\kappa\) still denotes the inclusion \[ U\times \text{Spec } \mathbb{C} [\tau]\hookrightarrow{\mathcal X}\times\text{Spec } \mathbb{C}[\tau]. \] This module is holonomic but not regular in general, so does not enter in the frame of mixed Hodge module theory. However, as \(\kappa_+ {\mathcal O}_U\) is regular on \({\mathcal X}\) (Grothendieck comparison theorem), \(\kappa_+ {\mathcal E}^{-\tau f}\) is regular along \(\tau=0\), so one may compute the vanishing cycles along \(\tau=0\) of its de Rham complex.
In \(\S 1\) some known facts concerning Fourier transform with parameters of sheaves are recalled. In §2 a Fourier transform with parameters of holonomic \({\mathcal D}\)-modules is introduced and a proof of a comparison theorem between both kinds of Fourier transforms is given. – The main result of \(\S 3\) is that one may apply the theory of Malgrange-Kashiwara filtration to Fourier transforms of regular holonomic \({\mathcal D}\)-modules to compute the nearby and vanishing cycles along \(\tau=0\). Section 4 is dedicated to Hodge theory of vanishing cycles at \(\tau=0\) of the Fourier transform relative to \(f\) of \(\kappa_+{\mathcal O}_U\) (and also the nearby cycles). A filtration naturally defined in terms of a natural filtration on \(\kappa_+ {\mathcal E}^{-\tau f}\) is put on this object [following the method developed by M. Saito, Publ. Res. Inst. Math. Sci. 24, No. 6, 849-995 (1988; Zbl 0691.14007) using the Malgrange-Kashiwara filtration]. In \(\S 5\) applications are given to the computation of the limit mixed Hodge structure of Steenbrink and Zucker in terms of the Fourier transform of the Gauss-Manin system of \(f\).

MSC:
14D07 Variation of Hodge structures (algebro-geometric aspects)
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
32G20 Period matrices, variation of Hodge structure; degenerations
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