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Théorie de Hodge des cycles évanescents. (Hodge theory of vanishing cycles). (French) Zbl 0538.14003

The aim of this paper is to study the variation of mixed Hodge structure arising from a projective morphism \(f: X\to D^*\) near the puncture of the disc \(D\) in \({\mathbb{C}}\). Namely to construct a limit mixed Hodge structure satisfying the condition given by P. Deligne in his article in Publ. Math., Inst. Hautes Étud. Sci. 52, 137-252 (1980; Zbl 0456.14014). We use the results on filtered mixed Hodge complex giving rise to spectral sequences of mixed Hodge structures [previous note, C. R. Acad. Sci., Paris, Sér. I 295, 669-672 (1982; Zbl 0511.14004)].
First we construct such a complex in the case where \(X\) is a normal crossing divisor in some ambiant space smooth over \(D^*\), and then we give the construction for a projective morphism. This paper contains the detailed proof with comments on previous work by W. Schmid, J. Steenbrink and H. Clemens in the case of smooth morphism f.
[For a short announcement of this paper see the following title.]

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14F40 de Rham cohomology and algebraic geometry
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References:

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