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The asymptotic behavior of a variation of polarized Hodge structure. (English) Zbl 0594.14012

The purpose of this paper is to give the asymptotic behavior of variation of polarized Hodge structures in the several-dimensional case. When S. Zucker [Ann. Math., II. Ser. 109, 415-476 (1979; Zbl 0446.14002)] proved that the cohomology groups of a variation of Hodge structure on the compact curve with finite singular points have also a Hodge structure, he was obliged to use the result by W. Schmid [Invent. Math. 22, 211-319 (1973; Zbl 0278.14003)] on the asymptotic behavior of the variation of Hodge structures at a singularity. In order to generalize Zucker’s result for varieties of higher dimension, the author first tried to generalize Schmid’s result to a higher dimensional case and got the result succcessfully in this paper. The same result is obtained by E. Cattani, A. Kaplan and W. Schmid, in a different way almost simultaneously.
Reviewer: M.Muro

MSC:

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

[1] P. Deligne, Theorie de Hodge I, II and III, Actes, Congres int. math., 1970, Tome 1, 425-430, Publ. Math. I.H.E.S., 40 (1972), 5-57 and ibid., 44 (1974), 5-77.
[2] P. Griffiths and W. Schmid, Locally homogeneous complex manifolds, Acta Math., 123 (1969), 253-302. · Zbl 0209.25701 · doi:10.1007/BF02392390
[3] W. Schmid, Variation of Hodge structure : the singularities of the period mappings, Inv. Math., 22 (1973), 211-319. · Zbl 0278.14003 · doi:10.1007/BF01389674
[4] J. Steenbrink and S. Zucker, Variation of mixed Hodge structure I, preprint. [Z] S. Zucker, Hodge theory with degenerating coefficients: LI cohomology in the Poin- car<§ metric, Annals of Math., 109 (1979), 415-476.
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