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Néron models and limits of Abel–Jacobi mappings

Part of: Families, fibrations (Co)homology theory Cycles and subschemes

Published online by Cambridge University Press:  02 February 2010

Mark Green ,
Phillip Griffiths  and
Matt Kerr
Mark Green
Affiliation:
Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095, USA (email: mlg@ipam.ucla.edu)
Phillip Griffiths
Affiliation:
Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA (email: pg@ias.edu)
Matt Kerr
Affiliation:
Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road/Durham DH1 3LE, UK (email: matthew.kerr@durham.ac.uk)
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Abstract

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We show that the limit of a one-parameter admissible normal function with no singularities lies in a non-classical sub-object of the limiting intermediate Jacobian. Using this, we construct a Hausdorff slit analytic space, with complex Lie group fibres, which ‘graphs’ such normal functions. For singular normal functions, an extension of the sub-object by a finite group leads to the Néron models. When the normal function comes from geometry, that is, a family of algebraic cycles on a semistably degenerating family of varieties, its limit may be interpreted via the Abel–Jacobi map on motivic cohomology of the singular fibre, hence via regulators on K-groups of its substrata. Two examples are worked out in detail, for families of 1-cycles on CY and abelian 3-folds, where this produces interesting arithmetic constraints on such limits. We also show how to compute the finite ‘singularity group’ in the geometric setting.

Keywords

Néron modelslit analytic spaceAbel–Jacobi mapadmissible normal functionvariation of Hodge structurelimit mixed Hodge structuremotivic cohomologyunipotent monodromysemistable reductionalgebraic cyclehigher Chow cycleCeresa cycleClemens–Schmid sequencepolarizationslit analytic space

MSC classification

Secondary: 14C25: Algebraic cycles 14C30: Transcendental methods, Hodge theory , Hodge conjecture 14C35: Applications of methods of algebraic $K$-theory 14D06: Fibrations, degenerations 14D07: Variation of Hodge structures 14F42: Motivic cohomology; motivic homotopy theory 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 2 , March 2010 , pp. 288 - 366
Copyright
Copyright © Foundation Compositio Mathematica 2010

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