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Hodge structure and holonomic systems. (English) Zbl 0629.14006

Let X be a compact complex variety of type C in the sense of Fujiki, and \(X_ 0\) a non-singular Zariski open subset of X. Let \((H_ Z,F)\) be a variation of Hodge structure on \(X_ 0\) of weight w. The well-known conjecture is that the cohomology group \(H^ n(X;^{\pi}H_ Q)\) has a Hodge structure of weight \(n+w.\)
Toward this direction, we gave the affirmative answer in Proc. Japan Acad., Ser. A 61, 164-167 (1985; Zbl 0576.14010; see also the preceding review) when X is a non-singular Kähler manifold and \(S=X\setminus X_ 0\) is a normally crossing hypersurface. It was also given by E. Cattani, A. Kaplan and W. Schmidt [Invent. Math. 87, 217-252 (1987; Zbl 0611.14006)] independently. But, the Hodge filtration of the cohomology groups was not given in an algebro-geometric way.
In this article we announce how to construct the Hodge filtration in an algebraic way.

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
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[1] Brylinski, J-L.: (Co)-homologie d’intersection et faisceaux pervers (Sem. Bourbaki, no. 585). Asterisque, 92/93, 129-157 (1982). · Zbl 0574.14017
[2] Cattani, E., A. Kaplan and W. Schmid: L2 and intersection cohomologies for a polarized variation of Hodge structure (preprint). · Zbl 0611.14006 · doi:10.1007/BF01389415
[3] Fujiki, A.: Closedness of the Douady spaces of compact Kahler spaces. Publ. RIMS, Kyoto Univ., 14, 1-52 (1978). · Zbl 0409.32016 · doi:10.2977/prims/1195189279
[4] Kashiwara, M. and T. Kawai: The Poincare lemma for a variation of polarized Hodge structure. Proc. Japan Acad., 61 A, 164-167 (1985). · Zbl 0576.14010 · doi:10.3792/pjaa.61.164
[5] Malgrange, B.: Ideals of Differentiable Functions. Oxford University Press (1966). · Zbl 0177.17902
[6] Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Inv. Math., 22, 211-319 (1973). · Zbl 0278.14003 · doi:10.1007/BF01389674
[7] Zucker, S.: Hodge theory with degenerating coefficients: L2 cohomology in the Poincare metric. Ann. of Math., 109, 415-476 (1979). JSTOR: · Zbl 0446.14002 · doi:10.2307/1971221
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