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Néron models and limits of Abel–Jacobi mappings

  • Mark Green (a1), Phillip Griffiths (a2) and Matt Kerr (a3)
Abstract
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.

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[2] F. Bardelli , Curves of genus three on a generic Abelian threefold and the nonfinite generation of the Griffiths group, in Arithmatic of complex manifolds (Erlanger, 1988), Lecture Notes in Mathematics, vol. 1399 (Springer, New York, 1989), 1026.

[8] J. Carlson , The geometry of the extension class of a mixed Hodge structure, in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proceedings of Symposia in Pure Mathematics, vol. 46 (American Mathematical Society, Providence, RI, 1987), 199222, part 2.

[14] P. Deligne , Equations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, vol. 163 (Springer, Berlin, 1970).

[18] W. Fulton , Intersection theory, second edition (Springer, New York, 1998).

[34] V. Kulikov and P. Kurchanov , Complex algebraic varieties: periods of integrals and Hodge structures, in Algebraic geometry III, Encyclopaedia of Mathematical Sciences, vol. 36 (Springer, Berlin, 1998), 1217.

[38] I. Nakamura , Relative compactification of the Néron model and its application, in Complex analysis and algebraic geometry (Iwanami Shoten, Tokyo, 1977), 207225.

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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