Variation of mixed Hodge structure. II (Q1820827)

In part I of this paper [\textit{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 \(\Sigma\neq 0)\) \((iii)\quad \ker \{H^{i+1}(Z,{\mathbb{C}})\to H^ 0(\bar S,R^{i+1}f_*{\mathbb{C}}\}\twoheadrightarrow H^ 1(\bar S,j_*{\mathbb{V}})\); \((iv)\quad H^ 2(\bar S,j_*{\mathbb{V}})\cong H^ 2(\bar S,R^ if_*{\mathbb{C}})\to H^{i+2}(Z,{\mathbb{C}})\).