Variation of mixed Hodge structure. II
type: Item
instance of: scholarly article
title: Variation of mixed Hodge structure. II
zbMATH Open document ID: 0615.14003
DOI: 10.1007/BF01388730
author: Steven Zucker
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143243
review text: 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 \(\Sigma\neq 0)\) \((iii)\quad \ker \{H^{i+1}(Z,{\mathbb{C}})\to H^ 0(\bar S,R^{i+1}f_*{\mathbb{C}}\}\twoheadrightarrow H^ 1(\bar S,j_*{\mathbb{V}})\); \((iv)\quad H^ 2(\bar S,j_*{\mathbb{V}})\cong H^ 2(\bar S,R^ if_*{\mathbb{C}})\to H^{i+2}(Z,{\mathbb{C}})\).
Mathematics Subject Classification ID: 14C30
zbMATH DE Number: 3995851
zbMATH Keywords: variation of mixed Hodge structure
zbMATH Keywords: mixed Hodge structure on the cohomology groups of a curve
MaRDI profile type: Publication
cites work: Q4094981
cites work: Stratifications de Whitney et théorème de Bertini-Sard
OpenAlex ID: W4244544029
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description: scientific article; zbMATH DE number 3995851