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SINGULARITIES OF THE BIEXTENSION METRIC FOR FAMILIES OF ABELIAN VARIETIES

Part of: Curves Families, fibrations Arithmetic algebraic geometry

Published online by Cambridge University Press:  23 July 2018

JOSÉ IGNACIO BURGOS GIL ,
DAVID HOLMES  and
ROBIN DE JONG
JOSÉ IGNACIO BURGOS GIL
Affiliation:
Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UCM3), Calle Nicolás Cabrera 15, Campus UAM, Cantoblanco, 28049 Madrid, Spain; burgos@icmat.es
DAVID HOLMES
Affiliation:
Mathematical Institute, Leiden University, PO Box 9512, 2300 RA Leiden, The Netherlands; holmesdst@math.leidenuniv.nl, rdejong@math.leidenuniv.nl
ROBIN DE JONG
Affiliation:
Mathematical Institute, Leiden University, PO Box 9512, 2300 RA Leiden, The Netherlands; holmesdst@math.leidenuniv.nl, rdejong@math.leidenuniv.nl

Abstract

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In this paper we study the singularities of the invariant metric of the Poincaré bundle over a family of abelian varieties and their duals over a base of arbitrary dimension. As an application of this study we prove the effectiveness of the height jump divisors for families of pointed abelian varieties. The effectiveness of the height jump divisor was conjectured by Hain in the more general case of variations of polarized Hodge structures of weight  $-1$ .

MSC classification

Primary: 14H10: Families, moduli (algebraic)
Secondary: 11G50: Heights 14D07: Variation of Hodge structures
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 6 , 2018 , e12
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2018

References

Amini, O., Bloch, S., Burgos Gil, J. and Fresán, J., ‘Feynman amplitudes and limits of heights’, Izv. Math. 80(5) (2016), 813–848.Google Scholar
Asakura, M., ‘Motives and algebraic de Rham cohomology’, inThe Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998), CRM Proc. Lecture Notes, 24 (American Mathematical Society, Providence, RI, 2000), 133–154.Google Scholar
Biesel, O., Holmes, D. and de Jong, R., ‘Néron models and the height jump divisor’, Trans. Amer. Math. Soc. 369 (2017), 8685–8723.Google Scholar
Birkenhake, C. and Lange, H., Complex Abelian Varieties, 2nd edn, Grundlehren der Mathematischen Wissenschaften, 302 (Springer, Berlin, 2004).Google Scholar
Boyd, S. and Vandenberghe, L., Convex Optimization, (Cambridge University Press, Cambridge, UK, 2009).Google Scholar
Brosnan, P. and Pearlstein, G., ‘Jumps in the Archimedean height’, Preprint, 2017, arXiv:1701.05527.Google Scholar
Burgos Gil, J. I., Holmes, D. and de Jong, R., ‘Positivity of the height jump divisor’, Int. Math. Res. Not. IMRN; doi:10.1093/imrn/rnx169.Google Scholar
Burgos Gil, J. I., Kramer, J. and Kühn, U., ‘The singularities of the invariant metric on the line bundle of Jacobi forms’, inRecent Advances in Hodge Theory, (eds. Kerr, M. and Pearlstein, G.) London Mathematical Society Lecture Note Series, 427 (Cambridge University Press, Cambridge, 2016), 45–77.Google Scholar
Cattani, E., ‘Mixed Hodge structures, compactifications and monodromy weight filtrations’, inTopics in Transcendental Algebraic Geometry, (ed. Griffiths, P.) Ann. of Math. Stud., 106 (Princeton University Press, Princeton, NJ, 1984), 75–100.Google Scholar
Deligne, P., ‘Le déterminant de la cohomologie’, inCurrent Trends in Arithmetical Algebraic Geometry (Arcata, CA, 1985), Contemporary Mathematics, 67 (American Mathematical Society, Providence, RI, 1987), 93–177.Google Scholar
Deligne, P., Equations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, 163 (Springer, Berlin, 1970).Google Scholar
Grothendieck, A., Groupes de monodromie en géométrie algébrique, Lecture Notes in Mathematics, 288 (Springer, Berlin, 1972).Google Scholar
Hain, R., ‘Biextensions and heights associated to curves of odd genus’, Duke Math. J. 61 (1990), 859–898.Google Scholar
Hain, R., ‘Normal functions and the geometry of moduli spaces of curves’, inHandbook of Moduli, Volume I, (eds. Farkas, G. and Morrison, I.) Advanced Lectures in Mathematics, XXIV (International Press, Boston, 2013).Google Scholar
Hayama, T. and Pearlstein, G., ‘Asymptotics of degenerations of mixed Hodge structures’, Adv. Math. 273 (2015), 380–420.Google Scholar
Holmes, D. and de Jong, R., ‘Asymptotics of the Néron height pairing’, Math. Res. Lett. 22(5) (2015), 1337–1371.Google Scholar
Kashiwara, M., ‘A study of variation of mixed Hodge structure’, Publ. RIMS, Kyoto Univ. 22 (1986), 991–1024.Google Scholar
Lear, D., ‘Extensions of normal functions and asymptotics of the height pairing’, PhD Thesis, University of Washington, 1990.Google Scholar
Pearlstein, G., ‘Variations of mixed Hodge structure, Higgs fields, and quantum cohomology’, Manuscripta Math. 102(3) (2000), 269–310.Google Scholar
Pearlstein, G., ‘SL2 -orbits and degenerations of mixed Hodge structure’, J. Differential Geom. 74 (2006), 1–67.Google Scholar
Pearlstein, G. and Peters, C., ‘Differential geometry of the mixed Hodge metric’, Commun. Anal. Geom. (to appear), Preprint, 2014, arXiv:1407.4082.Google Scholar
Peters, C. and Steenbrink, J., ‘Mixed Hodge structures’, inErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A Series of Modern Surveys in Mathematics, 52 (Springer, Berlin, 2008).Google Scholar
Rockafellar, R. T., Convex Analysis, Princeton Mathematical Series, 28 (Princeton University Press, Princeton, 1970).CrossRefGoogle Scholar
Saito, M., ‘Modules de Hodge polarisables’, Publ. Res. Inst. Math. Sci. 24 (1988), 849–995.CrossRefGoogle Scholar
Saito, M., ‘Mixed Hodge modules’, Publ. Res. Inst. Math. Sci. 26 (1990), 221–333.Google Scholar
Steenbrink, J. and Zucker, S., ‘Variation of mixed Hodge structure. I’, Invent. Math. 80 (1985), 489–542.Google Scholar