Hodge structure and holonomic systems
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 \textit{E. Cattani}, \textit{A. Kaplan} and \textit{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.
Hodge filtration of the cohomology groups
scientific article; zbMATH DE number 4023449