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Néron models and limits of Abel–Jacobi mappings

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

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: [email protected])
Phillip Griffiths
Affiliation:
Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA (email: [email protected])
Matt Kerr
Affiliation:
Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road/Durham DH1 3LE, UK (email: [email protected])
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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

References

[1]Asakura, M., A criterion of exactness of the Clemens–Schmid sequences arising from semi-stable families of open curves, Osaka J. Math. 40 (2003), 977–980.Google Scholar
[2]Bardelli, F., 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), 10–26.CrossRefGoogle Scholar
[3]Bloch, S., Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, CRM Monograph, vol. 11 (American Mathematical Society, Providence, RI, 2000).Google Scholar
[4]Boasch, S., Lütkebohmert, W. and Raynaud, M., Néron models (Springer, Berlin, 1944).Google Scholar
[5]Brosnan, P. and Pearlstein, G. J., The zero locus of an admissible normal function, Ann. of Math. (2) 170 (2009), 883–897.CrossRefGoogle Scholar
[6]Candelas, P., de la Ossa, X., Green, P. and Parkes, L., A pair of Calabai–Yau manifolds as an exactly solvable superconformal theory, in Essays on mirror manifolds (International Press, Hong Kong, 1992), 31–95.Google Scholar
[7]Carlson, J., Extensions of mixed Hodge structure, in Journées de Géometrie Algébrique d’Angers (Angers, 1979) (Sijthoff & Noordhoff, Alphen an den Rijn, The Netherlands, 1980), 107–127.Google Scholar
[8]Carlson, J., 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), 199–222, part 2.Google Scholar
[9]Carlson, J., Müller-Stach, S. and Peters, C., Period mappings and period domains, Stud. Adv. Math. 85 (2003).Google Scholar
[10]Ceresa, G., C is not algebraically equivalent to C − in its Jacobian, Ann. of Math. (2) 117 (1983), 285–291.CrossRefGoogle Scholar
[11]Clemens, H., Degeneration of Kähler manifolds, Duke Math. J. 44 (1977), 215–290.CrossRefGoogle Scholar
[12]Clemens, H., ‘The Néron model for families of intermediate Jacobians acquiring algebraic’ singularities, Publ. Inst. Hautes Études Sci. 58 (1983), 5–18.Google Scholar
[13]Collino, A., Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J. Algebraic Geom. 6 (1997), 393–415.Google Scholar
[14]Deligne, P., Equations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, vol. 163 (Springer, Berlin, 1970).CrossRefGoogle Scholar
[15]Doran, C. and Kerr, M., Algebraic K-theory of toric hypersurfaces, Preprint (2008), math.AG/08094669.Google Scholar
[16]Doran, C. and Morgan, J., Mirror symmetry and integral variations of Hodge structure underlying one parameter Yau threefolds, in Mirror symmetry V, AMS/IP Studies in Advanced Mathematics, vol. 28 (American Mathematical Society, Providence, RI, 2006).Google Scholar
[17]El Zein, F. and Zucker, S., Extendability of normal functions associated to algebraic cycles, in Topics in transcendental algebraic geometry, Annals of Mathematics Studies (Princeton University Press, Princeton, NJ, 1984), 269–288.CrossRefGoogle Scholar
[18]Fulton, W., Intersection theory, second edition (Springer, New York, 1998).CrossRefGoogle Scholar
[19]Green, M. and Griffiths, P., Algebraic cycles and singularities of normal functions, II, in Inspired by S. S. Chern: a memorial volume in honor of a great mathematian, Nanki Tracts in Mathematics, vol. 11, ed. Griffiths, (World Scientific, Singapore, 2006).Google Scholar
[20]Green, M. and Griffiths, P., Algebraic cycles and singularities of normal functions, in Algebraic cycles and motives, London Mathematical Society Lecture Note Series, vol. 343, eds Nagel, and Peters, (Cambridge University Press, Cambridge, 2007).Google Scholar
[21]Griffiths, P., On the periods of certain rational integrals II, Ann. of Math. (2) 90 (1969), 496–541.CrossRefGoogle Scholar
[22]Griffiths, P. and Schmid, W., Recent developments in Hodge theory, in Discrete subgroups of Lie groups and applications to moduli (Oxford University Press, Oxford, 1975), 31–128.Google Scholar
[23]Hardt, R., Slicing and intersection theory for chains associated with real analytic varieties, Acta Math. 129 (1972), 75–136.CrossRefGoogle Scholar
[24]Kashiwara, M., A study of variation of mixed Hodge structure, Publ. Res. Inst. Math. Sci. 22 (1986), 991–1024.CrossRefGoogle Scholar
[25]Kato, K. and Usui, S., Classifying spaces of degenerating polarized Hodge structures, Ann. Math. Stud., to appear.Google Scholar
[26]Katz, S., Degenerations of quintic threefolds and their lines, Duke Math. J. 50 (1983), 1127–1135.CrossRefGoogle Scholar
[27]Katz, S., Lines on complete intersection threefolds with K=0, Math. Z. 191 (1986), 293–296.CrossRefGoogle Scholar
[28]Kerr, M. and Lewis, J., The Abel–Jacobi map for higher Chow groups, II, Invent. Math. 170 (2007), 355–420.CrossRefGoogle Scholar
[29]Kerr, M., Lewis, J. and Müller-Stach, S., The Abel–Jacobi map for higher Chow groups, Compositio Math. 142 (2006), 274–396.CrossRefGoogle Scholar
[30]King, J., The currents defined by analytic varieties, Acta Math. 127 (1971), 185–220.CrossRefGoogle Scholar
[31]King, J., Global residues and intersections on a complex manifold, Trans. Amer. Math. Soc. 192 (1974), 163–199.CrossRefGoogle Scholar
[32]King, J., Log complexes of currents and functorial properties of the Abel–Jacobi map, Duke Math. J. 50 (1983), 1–53.CrossRefGoogle Scholar
[33]Kodaira, K., On compact analytic surfaces II, Ann. of Math. (2) 77 (1968), 563–626.CrossRefGoogle Scholar
[34]Kulikov, V. and Kurchanov, P., Complex algebraic varieties: periods of integrals and Hodge structures, in Algebraic geometry III, Encyclopaedia of Mathematical Sciences, vol. 36 (Springer, Berlin, 1998), 1–217.CrossRefGoogle Scholar
[35]Lian, B., Todorov, A. and Yau, S.-T., Maximal unipoint monodromy for complete intersection CY manifolds, Amer. J. Math. 127 (2005), 1–50.CrossRefGoogle Scholar
[36]Morrison, D., The Clemens–Schmid exact sequence and applications, in Topics in transcendental algebraic geometry, Annals of Mathematics Studies (Princeton University Press, Princeton, NJ, 1984), 101–119.CrossRefGoogle Scholar
[37]Morrison, D., Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians, J. Amer. Math. Soc. 6 (1993), 223–247.CrossRefGoogle Scholar
[38]Nakamura, I., Relative compactification of the Néron model and its application, in Complex analysis and algebraic geometry (Iwanami Shoten, Tokyo, 1977), 207–225.CrossRefGoogle Scholar
[39]Namikawa, Y., Toroidal degeneration of abelian varieties, II, Math. Ann. 245 (1979), 117–150.Google Scholar
[40]Pearlstein, G., Degenerations of mixed Hodge structure, Duke Math. J. 110 (2001), 217–251.CrossRefGoogle Scholar
[41]Pearlstein, G., SL2-orbits and degenerations of mixed Hodge structure, J. Differential Geom. 74 (2006), 1–67.CrossRefGoogle Scholar
[42]Pearlstein, G., Brosnan, P., Nie, Z. and Fang, H., Singularities and normal functions, Invent. Math. 177 (2009), 599–629.Google Scholar
[43]Saito, M., Admissible normal functions, J. Algebraic Geom. 5 (1996), 235–276.Google Scholar
[44]Schmid, W., Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319.CrossRefGoogle Scholar
[45]Schoen, C., Algebraic cycles on certain desingularized nodal hypersurfaces, Math. Ann. 270 (1985), 17–27.CrossRefGoogle Scholar
[46]Steenbrink, J., Limits of Hodge structures, Invent. Math. 31 (1976), 229–257.CrossRefGoogle Scholar
[47]Steenbrink, J. and Zucker, S., Variations of mixed Hodge structure I, Invent. Math. 80 (1985), 489–582.CrossRefGoogle Scholar
[48]Zucker, S., Generalized intermediate Jacobians and the theorems on normal functions, Invent. Math. 33 (1976), 185–222.CrossRefGoogle Scholar
[49]Zucker, S., Hodge theory with degenerating coefficients: L 2-cohomology in the Poincaré metric, Ann. of Math. (2) 109 (1979), 425–476.CrossRefGoogle Scholar
[50]Zucker, S., Degeneration of Hodge bundles (after Steenbrink), in Topics in transcendental algebraic geometry, Annals of Mathematical Studies, vol. 106 (Princeton University Press, Princeton, NJ, 1984), 121–141.CrossRefGoogle Scholar