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Gross–Schoen cycles and dualising sheaves

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References

  1. Beauville, A.: Quelques remarques sur la transformation de Fourier dans l’anneau de Chow d’une variété abélienne. In: Algebraic Geometry, Tokyo/Kyoto, 1982. Lecture Notes in Math., vol. 1016, pp. 238–260. Springer, Berlin (1983)

    Chapter  Google Scholar 

  2. Beauville, A.: Sur l’anneau de Chow d’une variété abélienne. Math. Ann. 273(4), 647–651 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  3. Beauville, A.: Algebraic cycles on Jacobian varieties. Compos. Math. 140(3), 683–688 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Beilinson, A.: Higher regulators and values of L-functions. J. Soviet Math. 30, 2036–2070 (1985)

    Article  MATH  Google Scholar 

  5. Beilinson, A.: Height pairing between algebraic cycles. In: Current Trends in Arithmetical Algebraic Geometry, Arcata, Calif., 1985. Contemp. Math., vol. 67, pp. 1–24. Amer. Math. Soc., Providence (1987)

    Google Scholar 

  6. Bloch, S.: Height pairings for algebraic cycles. J. Pure Appl. Algebra 34(2–3), 119–145 (1984). Proceedings of the Luminy Conference on Algebraic K-theory, Luminy, 1983

    Article  MATH  MathSciNet  Google Scholar 

  7. Bost, J.-B., Mestre, J.-F., Moret-Bailly, L.: Sur le calcul explicite des “classes de Chern” des surfaces arithmétiques de genre 2. Astérisque 183, 69–105 (1990). Séminaire sur les Pinceaux de Courbes Elliptiques, Paris, 1988

    MathSciNet  Google Scholar 

  8. Ceresa, G.: C is not algebraically equivalent to C in its Jacobian. Ann. Math. (2) 117(2), 285–291 (1983)

    Article  MathSciNet  Google Scholar 

  9. Chinburg, T., Rumely, R.: The capacity pairing. J. Reine Angew. Math. 434, 1–44 (1993)

    MATH  MathSciNet  Google Scholar 

  10. Cornalba, M., Harris, J.: Divisor classes associated to families of stable varieties, with applications to the moduli space of curves. Ann. Sci. École Norm. Sup. (4) 21(3), 455–475 (1988)

    MATH  MathSciNet  Google Scholar 

  11. Deligne, P.: Le déterminant de la cohomologie. In: Current Trends in Arithmetical Algebraic Geometry, Arcata, Calif., 1985. Contemp. Math., vol. 67, pp. 93–177. Amer. Math. Soc., Providence (1987)

    Google Scholar 

  12. Deligne, P.: Les constantes des équations fonctionnelles des fonctions L. In: Modular functions of one variable, II, Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972. Lecture Notes in Math., vol. 349, pp. 501–597. Springer, Berlin (1973)

    Chapter  Google Scholar 

  13. Elkies, N.D.: ABC implies Mordell. Int. Math. Res. Not. 7, 99–109 (1991)

    Article  MathSciNet  Google Scholar 

  14. Faber, X.: The geometric Bogomolov conjecture for small genus curves. Exp. Math., to appear (arXiv:0803.0855v2)

  15. Faltings, G.: Calculus on arithmetic surfaces. Ann. of Math. (2) 119(2), 387–424 (1984)

    Article  MathSciNet  Google Scholar 

  16. Faltings, G.: Lectures on the Arithmetic Riemann–Roch theorem. Ann. of Math. Stud., vol. 127. Princeton University Press, Princeton (1992). x+100 pp. Notes taken by Shouwu Zhang

    MATH  Google Scholar 

  17. Fulton, W.: Intersection Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, vol. 2. Springer, Berlin (1984). xi+470 pp.

    MATH  Google Scholar 

  18. Gillet, H., Soulé, C.: Arithmetic intersection theory. Inst. Hautes Études Sci. Publ. Math. 72, 93–174 (1990)

    Article  MATH  Google Scholar 

  19. Gillet, H., Soulé, C.: Characteristic classes for algebraic vector bundles with Hermitian metric. I. Ann. of Math. (2) 131(1), 163–203 (1990)

    Article  MathSciNet  Google Scholar 

  20. Gillet, H., Soulé, C.: Characteristic classes for algebraic vector bundles with Hermitian metric. II. Ann. of Math. (2) 131(2), 205–238 (1990)

    Article  MathSciNet  Google Scholar 

  21. Gillet, H., Soulé, C.: Arithmetic analogs of the standard conjectures. In: Motives, Seattle, WA, 1991. Proc. Sympos. Pure Math., vol. 55, pp. 129–140. Amer. Math. Soc., Providence (1994). Part 1

    Google Scholar 

  22. Gillet, H., Soulé, C.: An arithmetic Riemann–Roch theorem. Invent. Math. 110(3), 473–543 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  23. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Pure Appl. Math. Wiley–Interscience, New York (1978). xii+813 pp.

    MATH  Google Scholar 

  24. Gross, B., Kudla, S.: Heights and the central critical values of triple product L-functions. Compos. Math. 81(2), 143–209 (1992)

    MATH  MathSciNet  Google Scholar 

  25. Gross, B.H., Schoen, C.: The modified diagonal cycle on the triple product of a pointed curve. Ann. Inst. Fourier (Grenoble) 45(3), 649–679 (1995)

    MATH  MathSciNet  Google Scholar 

  26. Gubler, W.: The Bogomolov conjecture for totally degenerate abelian varieties. Invent. Math. 169(2), 377–400 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Künnemann, K.: Height pairings for algebraic cycles on abelian varieties. Ann. Sci. École Norm. Sup. (4) 34(4), 503–523 (2001)

    MATH  Google Scholar 

  28. Moret-Bailly, L.: Hauteurs et classes de Chern sur les surfaces arithméétiques. Astérisque 183, 37–58 (1990). Séminaire sur les Pinceaux de Courbes Elliptiques, Paris, 1988

    MathSciNet  Google Scholar 

  29. Moriwaki, A.: Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves. J. Amer. Math. Soc. 11(3), 569–600 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  30. Parshin, A.N.: The Bogomolov–Miyaoka–Yau inequality for the arithmetical surfaces and its applications. In: Séminaire de théorie des nombres, Paris, 1986–1987. Progr. Math., vol. 75, pp. 299–312. Birkhäuser, Boston (1988)

    Google Scholar 

  31. Raynaud, M.: Flat modules in algebraic geometry. Compos. Math. 24, 11–31 (1972)

    MATH  MathSciNet  Google Scholar 

  32. Szpiro, L.: Propriétés numériques du faisceau daulisant relatif. Astérisque 86, 44–78 (1981). Séminaire sur les pinceaux de courbes de genre au moint deaux

    MATH  Google Scholar 

  33. Tate, J.: Number theoretic background. In: Automorphic Forms, Representations and L-functions, Oregon State Univ., Corvallis, 1977. Proc. Sympos. Pure Math., vol. XXXIII, pp. 3–26. Amer. Math. Soc., Providence (1979). Part 2

    Google Scholar 

  34. Tate, J.: Algebraic cycles and poles of zeta functions. In: Arithmetical Algebraic Geometry. Proc. Conf. Purdue Univ., 1963, pp. 93–110. Harper, New York (1965)

    Google Scholar 

  35. Tate, J.: Conjectures on algebraic cycles in l-adic cohomology. In: Motives, Seattle, WA, 1991. Proc. Sympos. Pure Math., vol. 55, pp. 71–83. Amer. Math. Soc., Providence (1994). Part 1

    Google Scholar 

  36. Ullmo, E.: Positivité et discrétion des points algébriques des courbes. Ann. of Math. (2) 147(1), 167–179 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  37. Xiao, G.: Fibered algebraic surfaces with low slope. Math. Ann. 276(3), 449–466 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  38. Yamaki, K.: Effective calculation of the geometric height and the Bogomolov conjecture for hyperelliptic curves over function fields. J. Math. Kyoto Univ. 48(2), 401–443 (2008)

    MATH  MathSciNet  Google Scholar 

  39. Zhang, S.-W.: Admissible pairing on a curve. Invent. Math. 112(1), 171–193 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  40. Zhang, S.-W.: Small points and adelic metrics. J. Algebraic Geom. 4(2), 281–300 (1995)

    MATH  MathSciNet  Google Scholar 

  41. Zhang, S.-W.: Equidistribution of small points on abelian varieties. Ann. of Math. (2) 147(1), 159–165 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  42. Zhang, W.: Private communication

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Zhang, SW. Gross–Schoen cycles and dualising sheaves. Invent. math. 179, 1–73 (2010). https://doi.org/10.1007/s00222-009-0209-3

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