Variation of mixed Hodge structure. I. (English) Zbl 0626.14007

Around 1970 Griffiths introduced the notion of variation of Hodge structure on a complex manifold, which is the axiomatization of the features possessed by a local system of cohomology associated to a family of Kähler manifolds, parametrized by S. Subsequently, this theory has been developed in three main directions, namely: \((1)\quad Singularities\) of the period mapping,\((2)\quad De Rham\) theoretic realization of the limit mixed Hodge structure in geometric case, and\((3)\quad Hodge\) theory with degenerating coefficients. Then it was natural and in fact necessary, to consider generalizations to the case of variation of mixed Hodge structure, corresponding in geometry to families of varieties that are singular or non-compact.
This fundamental paper constitutes the first part of an attempt to construct a satisfactory theory for the mixed case, such that all the three directions stated above generalize. This answers in particular a question posed by P. Deligne [Publ. Math., Inst. Hautes Étud. Sci. 52, 137-252 (1980; Zbl 0456.14014)].
The following problem is also considered: if a given variation of mixed Hodge structure comes from a family of varieties \(f: Z\to S,\) to what extent the mixed Hodge structure for \(H^{\bullet}(S,{\mathbb{V}})\) is compatible with that of \(H^{\bullet}(Z)\). The answer turns out to be affirmative but difficult to prove. Finally, it is to be mentioned that recent contributions toward the theory of variation of mixed Hodge structure have been also brought by several people such as Usui, El Zein, Du Bois, Guillén, Navarro, Aznar and Puerta.
[For part II see Invent. Math. 80, 543-565 (1985; Zbl 0615.14003).]
Reviewer: L.Bădescu


14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
32G20 Period matrices, variation of Hodge structure; degenerations
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
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