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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:

[1] C. H. CLEMENS , Degeneration of Köhler Manifolds (Duke Math. Journal, vol. 44, 1977 , p. 215-290). Article | MR 444662 | Zbl 0353.14005 · Zbl 0353.14005 · doi:10.1215/S0012-7094-77-04410-6
[2] P. DELIGNE , Théorie de Hodge II (Publ. Math. I.H.E.S., vol. 40, 1972 , p. 5-57). Numdam | MR 498551 | Zbl 0219.14007 · Zbl 0219.14007 · doi:10.1007/BF02684692
[3] P. DELIGNE , Théorie de Hodge III (Publ. I.H.E.S., vol. 44, 1975 , p. 6-77). Numdam | MR 498552 | Zbl 0237.14003 · Zbl 0237.14003 · doi:10.1007/BF02685881
[4] P. DELIGNE , Weil II (Publ. Math. I.H.E.S., n^\circ 52, 1980 ). Numdam | Zbl 0456.14014 · Zbl 0456.14014 · doi:10.1007/BF02684780
[5] P. DELIGNE , Équations différentielles à points singuliers réguliers (Lecture Notes in Math., n^\circ 163, Springer, 1970 ). MR 417174 | Zbl 0244.14004 · Zbl 0244.14004 · doi:10.1007/BFb0061194
[6] P. DELIGNE et N. KATZ , Groupes de Monodromie en Géométrie algébrique (Springer Lecture Notes in Math., n^\circ 340, 1973 ). MR 354657 | Zbl 0258.00005 · Zbl 0258.00005
[7] F. ELZEIN , Complexes de Hodge mixtes filtrés (C. R. Acad. Sc., Paris, t. 295, série I, 1982 , p. 669-672). Zbl 0511.14004 · Zbl 0511.14004
[8] F. ELZEIN , Dégénérescence diagonale I et II (C. R. Acad. Sc., Paris, t. 292 et 296, série I, 1983 , p. 51-54 et p. 199-202). Zbl 0538.14004 · Zbl 0538.14004
[9] F. ELZEIN , Structures de Hodge mixtes (International conference in Math., Ryadh, 1982 ). Ed. Y. Al-Khamees. MR 609764 · Zbl 0468.14002
[10] F. ELZEIN , Mixed Hodge Structures (Transactions of the Amer. Math. Soc., vol. 275, 1983 ). MR 678337 | Zbl 0511.14003 · Zbl 0511.14003 · doi:10.2307/1999006
[11] P. GRIFFITHS , Periods of Integrals (Bull. of the Amer. Math. Soc., vol. 76, 1970 ). MR 258824 · Zbl 0214.19802
[12] P. GRIFFITHS et J. HARRIS , Principles of algebraic geometry , Wiley, New York. MR 1288523 | Zbl 0836.14001 · Zbl 0836.14001
[13] P. GRIFFITHS et W. SCHMID , Recent developments in Hodge theory (Discrete subgroups of Lie Groups, Bombay Colloquium, Oxford University press, 1973 ). MR 419850 | Zbl 0355.14003 · Zbl 0355.14003
[14] N. KATZ , Algebraic solutions of differential equations (Inv. Math., vol. 18, 1972 , p. 1-118). MR 337959 | Zbl 0278.14004 · Zbl 0278.14004 · doi:10.1007/BF01389714
[15] W. SCHMID , Variations of Hodge structure (Inventiones Math., vol. 22, 1973 , p. 211-319). MR 382272 | Zbl 0278.14003 · Zbl 0278.14003 · doi:10.1007/BF01389674
[16] J. STEENBRINK , Limits of Hodge structures (Inventiones Math., vol. 31, 1976 , p. 229-257). MR 429885 | Zbl 0303.14002 · Zbl 0303.14002 · doi:10.1007/BF01403146
[17] A. OGUS , Formal Hodge filtration (In. Math., vol. 31, 1976 , p. 193-228). MR 401765 | Zbl 0339.14004 · Zbl 0339.14004 · doi:10.1007/BF01403145
[18] J. L. VERDIER , Astérisque , n^\circ 36, 1976 . MR 481096 | Zbl 0346.14005 · Zbl 0346.14005
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