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Normal functions and the height of Gross-Schoen cycles

Published online by Cambridge University Press:  11 January 2016

Robin De Jong
Robin De Jong*
Affiliation:
Mathematical Institute University of Leiden2300 RA Leiden TheNetherlandsrdejong@math.leidenuniv.nl
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Abstract

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We prove a variant of a formula due to Zhang relating the Beilinson– Bloch height of the Gross–Schoen cycle on a pointed curve with the self-intersection of its relative dualizing sheaf. In our approach, the height of the Gross–Schoen cycle occurs as the degree of a suitable Bloch line bundle. We show that the Chern form of this line bundle is nonnegative, and we calculate its class in the Picard group of the moduli space of pointed stable curves of compact type. The basic tools are normal functions and biextensions associated to the cohomology of the universal Jacobian.

Keywords

14G4014C2514D06
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 214 , June 2014 , pp. 53 - 77
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[1] Arakelov, S. Y., An intersection theory for divisors on an arithmetic surface (in Russian), Izv. Akad. Nauk. SSSR Ser. Mat. 38 1974, 1179–1192h. MR 0472815.Google Scholar
[2] Arbarello, E. and Cornalba, M., The Picard groups of the moduli spaces of curves, Topology 26 1987, 153–171. MR 0895568. DOI 10.1016/0040-9383 (87)90056-5.Google Scholar
[3] Beauville, A., “Quelques remarques sur la transformation de Fourier dans l’anneau de Chow d’une variété abelienne” in Algebraic Geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math. 1016, Springer, Berlin, 1983, 238–260. MR 0726428. DOI 10.1007/ BFb0099965.Google Scholar
[4] Beilinson, A., “Height pairing between algebraic cycles” in Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985), Contemp. Math. 67, Amer. Math. Soc, Providence, 1987, 1–24. MR 0902590. DOI 10.1090/conm/067/902590.Google Scholar
[5] Bloch, S., “Height pairings for algebraic cycles” in Proceedings of the Luminy Conference on Algebraic K-theory (Luminy, 1983), J. Pure Appl. Algebra 34, 1984, 119–145. MR 0772054. DOI 10.1016/0022-4049(84)90032-X.CrossRefGoogle Scholar
[6] Bloch, S., “Cycles and biextensions” in Algebraic K-theory and Algebraic Number The ory (Honolulu, 1987), Contemp. Math. 83, Amer. Math. Soc, Providence, 1989, 19–30. MR 0991974. DOI 10.1090/conm/083/991974.Google Scholar
[7] Bost, J.-B., Gillet, H., and Soulé, C., Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7 1994, 903–1027. MR 1260106. DOI 10.2307/2152736.Google Scholar
[8] Colombo, E. and Geemen, B. van, Note on curves in a Jacobian, Compos. Math. 88 1993, 333–353. MR 1241954.Google Scholar
[9] Jong, R. de, Second variation of Zhang’s λ-invariant on the moduli space of curves, Amer. J. Math. 135 2013, 275–290. MR 3022965. DOI 10.1353/ajm.2013.0008.Google Scholar
[10] Deligne, P., “Le déterminant de la cohomologie” in Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985), Contemp. Math. 67, Amer. Math. Soc, Providence, 1987, 93–177. MR 0902592. DOI 10.1090/conm/067/902592.CrossRefGoogle Scholar
[11] Deninger, C. and Murre, J., Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math. 422 1991, 201–219. MR 1133323.Google Scholar
[12] Faltings, G., Calculus on arithmetic surfaces, Ann. of Math. (2) 119 1984, 387–424. MR 0740897. DOI 10.2307/2007043.Google Scholar
[13] Gross, B. H. and Schoen, C., The modified diagonal cycle on the triple product of a pointed curve, Ann. Inst. Fourier (Grenoble) 45 1995, 649–679. MR 1340948.Google Scholar
[14] Hain, R., Biextensions and heights associated to curves of odd genus, Duke Math. J. 61 1990, 859–898. MR 1084463. DOI 10.1215/S0012–7094-90–06133-2.Google Scholar
[15] Hain, R., “Normal functions and the geometry of moduli spaces of curves” in Handbook of Moduli, Adv. Lectures Math. 24, International Press, Boston, 2013, 527–556.Google Scholar
[16] Hain, R. and Reed, D., Geometric proofs of some results of Morita, J. Algebraic Geom. 10 2001, 199–217. MR 1811554.Google Scholar
[17] Hain, R. and Reed, D., On the Arakelov geometry of moduli spaces of curves, J. Differential Geom. 67 2004, 195–228. MR 2153077.Google Scholar
[18] Harris, B., Harmonic volumes, Acta Math. 150 1983, 91–123. MR 0697609. DOI 10. 1007/BF02392968.CrossRefGoogle Scholar
[19] Iversen, B., Cohomology of Sheaves, Universitext, Springer, Berlin, 1986. MR 0842190. DOI 10.1007/978–3-642–82783-9.Google Scholar
[20] Kausz, I., A discriminant and an upper bound for ω2 for hyperelliptic arithmetic surfaces, Compos. Math. 115 1999, 37–69. MR 1671741. DOI 10.1023/A: 1000580901251.Google Scholar
[21] Meyer, O., Über Biextensionen und Höhenpaarungen algebraischer Zykel, Ph.D. dissertation, University of Regensburg, Regensburg, Germany, 2003.Google Scholar
[22] Moret-Bailly, L., “Métriques permises” in Seminar in Arithmetic Bundles: The Mordell Conjecture (Paris, 1983/84), Astérisque 127, Soc. Math. France, Paris, 1985, 29–87. MR 0801918.Google Scholar
[23] Moret-Bailly, L., La formule de Noether pour les surfaces arithmétiques, Invent. Math. 98 1989, 491–498. MR 1022303. DOI 10.1007/BF01393833.Google Scholar
[24] Mu¨ller-Stach, S., C∗-extensions of tori, higher Chow groups and applications to incidence equivalence relations for algebraic cycles, K-Theory 9 1995, 395–406. MR 1351945. DOI 10.1007/BF00961471.Google Scholar
[25] Pulte, M. J., The fundamental group of a Riemann surface: Mixed Hodge structures and algebraic cycles, Duke Math. J. 57 1988, 721–760. MR 0975119. DOI 10.1215/ S0012–7094-88–05732-8.Google Scholar
[26] Seibold, M., Bierweiterungen für algebraische Zykel und Poincarébundel, Ph.D. dissertation, University of Regensburg, Regensburg, Germany, 2007.Google Scholar
[27] Szamuely, T., Galois Groups and Fundamental Groups, Cambridge Stud. Adv. Math. 117, Cambridge University Press, Cambridge, 2009. MR 2548205. DOI 10.1017/ CBO9780511627064.Google Scholar
[28] Zhang, S.-W., Admissible pairing on a curve, Invent. Math. 112 1993, 171–193. MR 1207481. DOI 10.1007/BF01232429.Google Scholar
[29] Zhang, S.-W., Gross–Schoen cycles and dualising sheaves, Invent. Math. 179 2010, 1–73. MR 2563759. DOI 10.1007/s00222–009-0209–3.CrossRefGoogle Scholar
[30] Zhang, S.-W., Positivity of heights of codimension 2 cycles over function field of charac-teristic 0, preprint, arXiv:1001.4788v1 [math.AG].Google Scholar