## Variation of mixed Hodge structure. II.(English)Zbl 0615.14003

In part I of this paper [J. Steenbrink and the author, ibid. 80, 489-542 (1985)] put a mixed Hodge structure on the cohomology groups of a curve with values in $${\mathbb{V}}$$, where $${\mathbb{V}}$$ is a graded-polarizable variation of mixed Hodge structures. In the paper under review the author considers the case where $${\mathbb{V}}$$ arises from geometry. Let $$f: Z\to \bar S$$ be a family of quasi-projective varieties over a smooth complete curve, let $$S=\bar S-\Sigma$$ be the set of regular values for f, let $$g: U\to S$$ be the restriction of f to S, and let $${\mathbb{V}}=R^ ig_*{\mathbb{C}}$$. Then the following are morphisms of mixed Hodge structure: $$(i)\quad \pi_ i: H^ i(U,{\mathbb{C}})\to H^ 0(S,{\mathbb{V}})\cong H^ 0(\bar S,j_*{\mathbb{V}})$$; $$(ii)\quad \ker \pi_{i+1}\twoheadrightarrow H^ 1(S,{\mathbb{V}})$$ (isomorphism if $$\Sigma\neq 0)$$ $$(iii)\quad \ker \{H^{i+1}(Z,{\mathbb{C}})\to H^ 0(\bar S,R^{i+1}f_*{\mathbb{C}}\}\twoheadrightarrow H^ 1(\bar S,j_*{\mathbb{V}})$$; $$(iv)\quad H^ 2(\bar S,j_*{\mathbb{V}})\cong H^ 2(\bar S,R^ if_*{\mathbb{C}})\to H^{i+2}(Z,{\mathbb{C}})$$.
Reviewer: J.A.Carlson

### MSC:

 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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### References:

 [1] Steenbrink, J.: Limits of Hodge structures and intermediate Jacobians. Thesis. University of Amsterdam, 1974 · Zbl 0329.14007 [2] Verdier, J.-L. Stratifications de Whitney et théorème de Bertini-Sard. Invent. Math.36, 295-312 (1976) · Zbl 0333.32010
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