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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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