Hodge theory with degenerating coefficients: \(L_2\) cohomology in the Poincaré metric
type: Item
instance of: scholarly article
title: Hodge theory with degenerating coefficients: \(L_2\) cohomology in the Poincaré metric
zbMATH Open document ID: 0446.14002
DOI: 10.2307/1971221
author: Steven Zucker
published in: Annals of Mathematics. Second Series
publication date: 1979-01-01T00:00:00Z
review text: Let \(\bar S\) be a smooth algebraic curve over \(\mathbb C\) and \(S\) an open subset of \(\bar S\). Let there be a locally constant system \(V\) on \(S\) of complex vector spaces underlying a polarizable variation of Hodge structures (over \(\mathbb R)\) of weight \(m\). Such a system is a Hodge theory with degenerating coefficients. The main result of the paper is that there exists a natural polarizable Hodge structure of weight \(m+i\) on \(H^i(\bar S, j_*V)\), \(j\colon S\to \bar S\) the inclusion. This Hodge structure is constructed by using \(L_2\)-differential forms with respect to a suitable metric on \(S\). The applications include: the author's theorem on normal functions relating to the Hodge conjecture [Invent. Math. 33, 185--222 (1976; Zbl. 329.14008)]; the Hodge structures of Shimura on the cohomology of Fuchsian groups. In the last section \(H^*(\bar S, j_*V)\) is compared with \(H^*(X,\mathbb C)\) for a smooth family \(f\colon X\to S\) of varieties giving rise to the local system \(V\). Sections 13, 14 deal with some associated mixed Hodge structures and cohomology with compact support.
reviewed by: G. Horrocks
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 32J25
zbMATH DE Number: 3697266
zbMATH Keywords: polarizable variation of Hodge structures
zbMATH Keywords: Hodge theory with degenerating coefficients
zbMATH Keywords: L2-differential forms
zbMATH Keywords: cohomology of Fuchsian groups
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971221
OpenAlex ID: W2328158498
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description: scientific article; zbMATH DE number 3697266