Compositio Mathematica

Research Article

Néron models and limits of Abel–Jacobi mappings

Mark Greena1, Phillip Griffithsa2 and Matt Kerra3

a1 Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095, USA (email: mlg@ipam.ucla.edu)

a2 Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA (email: pg@ias.edu)

a3 Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road/Durham DH1 3LE, UK (email: matthew.kerr@durham.ac.uk)

Abstract

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.

(Received August 15 2007)

(Accepted January 15 2009)

(Online publication February 02 2010)

2000 Mathematics Subject Classification

  • 14C25 (primary);
  • 14C30;
  • 14C35;
  • 14D06;
  • 14D07;
  • 14F42;
  • 14F43 (secondary)

Keywords

  • Néron model;
  • slit analytic space;
  • Abel–Jacobi map;
  • admissible normal function;
  • variation of Hodge structure;
  • limit mixed Hodge structure;
  • motivic cohomology;
  • unipotent monodromy;
  • semistable reduction;
  • algebraic cycle;
  • higher Chow cycle;
  • Ceresa cycle;
  • Clemens–Schmid sequence;
  • polarization;
  • slit analytic space