Complex analytic Néron models for arbitrary families of intermediate Jacobians
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 \textit{K. Kato} et al. [Proc. Japan Acad., Ser. A 86, No. 8, 143--148 (2010; Zbl 1209.14009)] and in [Proc. Japan Acad., Ser. A 87, No. 9, 167--172 (2011; Zbl 1252.14010)] in the category of log manifolds. In the case of admissible normal functions with torsion singularities the author actually constructs an analytic Néron model; which is argued to be the most general setting on which a Néron model exists as an analytic or even as a Hausdorff space at that time. The author's construction is based on unpublished work by H. Clemens on the family of hyperplane sections of a smooth projective variety and uses filtered \(\mathcal D\)-modules and on \textit{M. Saito}'s theory of mixed Hodge modules in [Publ. Res. Inst. Math. Sci. 26, No. 2, 221--333 (1990; Zbl 0727.14004)]. It has the advantage of being functorial and of having no assumptions on the singularities of \(D=\overline X\setminus X\). It has the advantage of being functorial and of having no assumptions on the singularities of \(D=\overline X\setminus X\).
scientific article; zbMATH DE number 6037976
scientific article