Variation of mixed Hodge structure arising from family of logarithmic deformations. II: Classifying space
type: Item
instance of: scholarly article
title: Variation of mixed Hodge structure arising from family of logarithmic deformations. II: Classifying space
zbMATH Open document ID: 0558.14005
DOI: 10.1215/S0012-7094-84-05137-8
author: Sampei Usui
published in: Duke Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: The present article is a continuation of part I [the author, Ann. Sci. Ec. Norm. Supér., IV. Sér. 16, 91-107 (1983; Zbl 0516.14006)]. Let \((H_{{\mathbb{Z}}},W,F,Q)\) be a gradedly polarized mixed Hodge structure of weight length \(=1\). The following is the main result: (1) The classifying space D of Hodge filtrations F, for fixed \(H_{{\mathbb{Z}}}\), W and Q, has a structure of homogeneous, holomorphic vector bundle over a product of Griffiths domains \((=\) the classifying space of Hodge structure). (2) D is acted properly discontinuously by \(G_{{\mathbb{Z}}}:=\{g\in GL(H_{{\mathbb{Z}}})| \quad g(W\bullet)=W\bullet \}.\) (3) D carries the extended horizontal subbundle \(T_ D^{eh}\) of the tangent bundle, which is compatible with the horizontal subbundle of Griffiths. (4) The period map associated to a variation of gradedly polarized mixed Hodge structure arising from geometry has extended horizontal local liftings. These are compatible with the Griffiths period map and its horizontal local liftings. Some results from hyperbolicity of horizontal subbundle are derived in the present situation, e.g., quasi-unipotency of local monodromy. - The present article is independently done but closely related with a preprint of \textit{J. Carlson}: ''The obstruction of splitting a mixed Hodge structure over the integers. I'', and the condition ''weight length \(=1''\) can be eliminated by a result of Carlson [cf. also ''Supplement...'' by \textit{M. Saito}, \textit{Y. Shimizu} and the author in Duke Math. J. 52, 529- 534 (1985)].
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 32G20
zbMATH DE Number: 3889692
zbMATH Keywords: extended horizontal subbundle
zbMATH Keywords: classifying space
zbMATH Keywords: polarized mixed Hodge structure
zbMATH Keywords: period map
MaRDI profile type: Publication
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full work available at URL: s0012-7094-84-05137-8
OpenAlex ID: W1482828651
description: scientific article; zbMATH DE number 3889692