Complex analytic Néron models for arbitrary families of intermediate Jacobians
type: Item
instance of: scholarly article
title: Complex analytic Néron models for arbitrary families of intermediate Jacobians
zbMATH Open document ID: 1299.14009
author: Christian Schnell
published in: Inventiones Mathematicae
publication date: 2012-05-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/0910.0662
review text: This paper is devoted to solving a question posed by P. Griffiths on the construction of Néron models for arbitrary families of compactified Jacobians. Namely, given a family of intermediate Jacobians (coming from a polarizable variation of Hodge structure \(\mathcal H\) of odd weight) defined over a Zariski open subset \(X\) of a complex manifold \(\overline X\), the author constructs an analytic space \(\overline J(\mathcal H)\) that naturally extends the family to all \(\overline X\) in such a way that admissible normal functions extend to sections of \(\overline J(\mathcal H)\). The space \(\overline J(\mathcal H)\) constructed in the paper has all the properties that the identity component of the Néron model is expected to satisfy, namely that ``its horizontal and holomorphic sections are precisely the admissible normal functions without singularities''. An actual Néron model for the family of intermediate Jacobians should then have other components satisfying the following properties: ``(1) Over each point of \(\overline X\) its fiber should be a countable union of complex Lie groups. (2) The components over a point \(x\in \overline X\setminus X\) where the variation degenerates should be indexed by a countable group, whose elements are the possible values for the \textit{singularity} at \(x\) of admissible normal functions...''. In the paper the author also shows that the graph of any admissible normal function has an analytic closure inside \(\overline J(\mathcal H)\). One consequence of this is a new proof for a conjecture by M. Green and P. Griffiths which predicts that given an admissible normal function \(\nu\) on an algebraic variety \(X\), its zero locus \(Z(\nu)\) is an algebraic subvariety of \(X\). Other solutions to Griffiths' problem have been given by \textit{P. Brosnan, G. Pearlstein} and \textit{M. Saito} in [``A generalization of the Neron models of Green, Griffiths and Kerr'', (2008). \url{arXiv:0809.5185}] as a topological space and by \text…
reviewed by: Margarida Melo
Mathematics Subject Classification ID: 14D07
Mathematics Subject Classification ID: 32G20
Mathematics Subject Classification ID: 14K30
zbMATH DE Number: 6037976
zbMATH Keywords: Néron models
zbMATH Keywords: intermediate Jacobians
zbMATH Keywords: variation of integral Hodge structure
MaRDI profile type: Publication
OpenAlex ID: W2155954281
arXiv ID: 0910.0662
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DOI: 10.1007/S00222-011-0341-8
description: scientific article; zbMATH DE number 6037976