×

Néron models and limits of Abel-Jacobi mappings. (English) Zbl 1195.14006

In this paper, one can find the construction of a slit analytic fibre space of complex Lie groups, according to the terminology of K. Kato and S. Usui [Classifying spaces of degenerating polarized Hodge structures.Annals of Mathematics Studies 169. Princeton, NJ: Princeton University Press. (2009; Zbl 1172.14002)], which graphs admissible normal functions with no singularities, in a similar way as the classical Néron model [S. Bosch, W. Lütkebohmert and M. Raynaud, Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 21. Berlin etc.: Springer-Verlag. (1990; Zbl 0705.14001)] graphs admissible normal functions arising from families of curves (the definition of admissible normal function can be found in: [M. Saito, J. Algebr. Geom. 5, No.2, 235–276 (1996; Zbl 0918.14018)].
The contents of this paper are best presented by the authors’ 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.”

MSC:

14C25 Algebraic cycles
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14D06 Fibrations, degenerations in algebraic geometry
14D07 Variation of Hodge structures (algebro-geometric aspects)
14F42 Motivic cohomology; motivic homotopy theory
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] doi:10.1007/BF01428801 · Zbl 0405.14020
[2] doi:10.1215/S0012-7094-01-11022-3 · Zbl 1092.14018
[4] doi:10.2307/2152798
[9] doi:10.1353/ajm.2005.0006 · Zbl 1067.14035
[10] doi:10.4007/annals.2009.170.883 · Zbl 1184.32004
[13] doi:10.2307/1970131 · Zbl 0118.15802
[15] doi:10.1215/S0012-7094-83-05001-9 · Zbl 0526.32011
[16] doi:10.1007/BFb0095965
[17] doi:10.2307/1996827 · Zbl 0301.32005
[19] doi:10.1007/BF02392053 · Zbl 0224.32008
[20] doi:10.1112/S0010437X05001867 · Zbl 1123.14006
[21] doi:10.1007/s00222-007-0066-x · Zbl 1139.14010
[22] doi:10.1007/BF01164033 · Zbl 0563.14020
[23] doi:10.1215/S0012-7094-83-05048-2 · Zbl 0616.14035
[24] doi:10.2977/prims/1195177264 · Zbl 0621.14007
[25] doi:10.1007/BF02392214 · Zbl 0234.32005
[27] doi:10.2307/1970747
[32] doi:10.2307/1971221 · Zbl 0446.14002
[34] doi:10.1007/BF01404203 · Zbl 0329.14008
[36] doi:10.1007/BF01388729 · Zbl 0626.14007
[37] doi:10.1007/BF01403146 · Zbl 0303.14002
[39] doi:10.1007/BF01455524 · Zbl 0533.14002
[41] doi:10.1007/BF01389674 · Zbl 0278.14003
[44] doi:10.1215/S0012-7094-77-04410-6 · Zbl 0353.14005
[45] doi:10.1007/s00222-009-0191-9 · Zbl 1174.14009
[46] doi:10.2307/2007078 · Zbl 0538.14024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.