Inventiones mathematicae

, Volume 110, Issue 1, pp 473–543 | Cite as

An arithmetic Riemann-Roch theorem

  • Henri Gillet
  • Christophe Soulé
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References

  1. [BFM] Baum, P., Fulton, W., MacPherson R.: Riemann-Roch for singular varieties. Publ. Math., Inst. Hautes Etud. Sci.45, 101 145 (1975)Google Scholar
  2. [BGI] Grothendieck, A., Berthelot, P., Illusie, L.: SGA6, Théorie des intersections et théorème de Riemann-Roch. (Lect. Notes Math., vol 225) Berlin Heidelberg New York: Springer 1971Google Scholar
  3. [B] Bismut, J.-M.: Superconnection currents and complex immersions. Invent. Math.99, 59–113 (1990)Google Scholar
  4. [BGS1] Bismut, J.-M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles I, II, III. Commun. Math. Phys.115, 49–78, 79–126, 301–351 (1988)Google Scholar
  5. [BGS2] Bismut, J.-M., Gillet, H. et Soulé, C.: Bott-Chern currents and complex immersions. Duke Math. J.60, 255–284 (1990)Google Scholar
  6. [BGS3] Bismut, J.-M., Gillet, H., Soulé, C.: Complex immersions and Arakelov geometry. In. Grothendieck Festschrift I, pp. 249–331. Boston Basel Stuttgart: Birkhäuser, 1990Google Scholar
  7. [BL] Bismut J.-M., Lebeau, G.: Complex immersions and Quillen metrics. Publ. Math., Inst. Hautes Étud. Sci. (to appear) (see also “Immersions complexes et métriques de Quillen”. C.R. Acad. Sci, Paris309, 487–491 (1989))Google Scholar
  8. [BV] Bismut, J.-M., Vasserot, E.: The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle. Commun. Math. Physics125, 355–367 (1989)Google Scholar
  9. [Bo] Bombieri, E.: The Mordell conjecture revisited. Ann. Scuola Mormale Superione Pisa Serie IV,17, 2, 615–640 (1990)Google Scholar
  10. [BC] Bott, R., Chern, S.S.: Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math.114, 71–112 (1968)Google Scholar
  11. [D] Deligne, P.: Le déterminant de la cohomologie. In: Ribet, K.A. (ed.) Current Trends in Arithmetical Algebraic Geometry. (Contemp. Math. vol. 67, pp. 93–178) Providence, RI: Am. Math. Soc. 1987Google Scholar
  12. [EGA1] Dieudonné, J. Grothendieck, A.: Éléments de Géométrie Algébrique I (Grund. Math. Wiss., vol. 166) Berlin Heidelberg New York: Springer 1971Google Scholar
  13. [EGA2] Dieudonné, J., Grothendieck, A.: Eléments de Géométrie Algébrique II. Publ. Math., Inst. Hautes Étud. Sci.8 (1961)Google Scholar
  14. [EGA4] Dieudonné, J., Grothendieck, A.: Éléments de Géométrie Algébrique IV. Publ. Math. Inst. Hautes Étud. Sci.20 (1964)Google Scholar
  15. [Do] Donaldson, S.K.: Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. 3,50, 1–26 (1985)Google Scholar
  16. [E] Elkik, R.: Métriques sur les fibrés d'intersection. Duke Math. J.61, 303–328 (1990)Google Scholar
  17. [F1] Faltings, G.: Calculus on arithmetic surfaces. Ann. Math.119, 387–424 (1984)Google Scholar
  18. [F2] Faltings, G.: Diophantine approximation on abelian varieties. Ann. Math. 2133, 549–576 (1991)Google Scholar
  19. [F3] Faltings, G.: Lectures on the arithmetic Riemann Roch theorem, notes by S. Zhang. Ann. Math. Studies 127 (1992)Google Scholar
  20. [Fr] Franke, J.: Riemann Roch in functorial form. (Preprint 1990)Google Scholar
  21. [Fu1] Fulton, W.: Rational equivalence on singular varieties. Publ. Math. Inst. Hautes Étud. Sci.45, 147–167 (1975)Google Scholar
  22. [Fu2] Fulton, W.: Intersection Theory. (Ergeb. Math. Grenzgeb., 3. Folge, Band 2) Berlin Heidelberg New York: Springer 1984Google Scholar
  23. [G1] Gillet, H.: Riemann-Roch for higher algebraic K-theory. Adv. Math.40, 203 289 (1981)Google Scholar
  24. [G2] Gillet, H.: Homological descent for the K-theory of coherent sheaves. In: Bak, A. (ed.) Algebraic K-Theory, Number Theory, Geometry and Analysis) (Lect. Notes Math., vol. 1046, pp. 80–104) Berlin Heidelberg New York: Springer 1984Google Scholar
  25. [G3] Gillet, H.: A Riemann-Roch theorem in arithmetic geometry. In. Proceedings International Congress of Mathematicians, Kyoto, 1990, 403 413, SpringerGoogle Scholar
  26. [GS1] Gillet, H., Soulé, C.: Intersection theory using Adams operations. Inven Math.90, 243–278 (1987)Google Scholar
  27. [GS2] Gillet, H. and Soulé, C.: Arithmetic intersection theory. Publ. Math., Inst. Hautes Étud. Sci.72, 94–174 (1990)Google Scholar
  28. [GS3] Gillet, H., Soulé, C.: Characteristic classes for algebraic vector bundles with Hermitian metrics, I, II. Ann. Math.131, 163–203 and 205 238 (1990)Google Scholar
  29. [GS4] Gillet, H., Soulé, C.: Analytic torsion and the arithmetic Todd genus, with an Appendix by D. Zagier. Topology30, n1, 21–54 (1991)Google Scholar
  30. [GS5] Gillet, H., Soulé, C.: On the number of lattice points in convex symmetric bodies and their duals, Isr. J. Math.74, 347 357 (1991)Google Scholar
  31. [GS6] Gillet, H. and Soulé, C.: Amplitude arithmétique. Note C.R. Acad. Sci. Paris. Sér. I307, 887–890 (1988)Google Scholar
  32. [GS7] Gillet, H., Soulé, C.: Un théorème de Riemann Roch Grothendieck arithmétique, Note CRAS ParisI, 309, 929 932 (1989)Google Scholar
  33. [GH] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Chichester: John Wiley and Sons 1978Google Scholar
  34. [H1] Hartshorne, R.: Ample vector bundles. Publ. Math. Inst. Hautes Étud. Sci.29, 63 94 (1966)Google Scholar
  35. [H2] Hartshorne, R.: Algebraic Geometry. (Grad. Texts Math., vol. 52) Berlin Heidelberg New York: Springer 1977Google Scholar
  36. [Hz] Hirzebruch, F.: Topological methods in algebraic geometry. (Grundl. Math., vol. 131) Berlin Heidelberg New York: Springer 1966Google Scholar
  37. [KM] Knudsen, F.F., Mumford, D.: The projectivity of the moduli space of stable curves I: Preliminaries on “det” and “div”. Math. Scand.39, 19 55 (1976)Google Scholar
  38. [L] Lafforgue, L.: Une version en géométrie diophantienne du “Lemme de l'indice”. Eco. Norm. Supér. (Preprint 1990)Google Scholar
  39. [Q1] Quillen, D.: Higher Algebraic K-Theory I. (Lect. Notes Math., vol. 341, pp. 85–147) Berlin Heidelberg New York: 1973Google Scholar
  40. [Q2] Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl.19, 31–34 (1985)Google Scholar
  41. [Q3] Quillen, D.: Superconnections and the Chern character. Topology24, 89 95 (1985)Google Scholar
  42. [RS] Ray, D.B., Singer, I.M.: Analytic torsion for complex manifolds. Ann. Math.98, 154–177 (1973)Google Scholar
  43. [S1] Soulé, C.: Opérations en K-théorie algébrique. Can. J. Math. 37, no. 3 (1985), 488–550Google Scholar
  44. [S2] Soulé, C.: Géométrie d'Arakelov des surfaces arithmétiques. In: Séminaire, Bourbaki 713. Astérisque 177–178, 327–343 (1989)Google Scholar
  45. [S3] Soulé, C.: Geométrie d'Arakelov et théorie des nombres transcendants, Astérisque 198-199-200, 355–373 (1991)Google Scholar
  46. [V] Vardi, I.: Determinant of Laplacians and multiple gamma functions. SIAM J. Math. Anal.19, 493–507 (1988)Google Scholar
  47. [Vo] Vojta, P.: Siegel's Theorem in the compact case. Ann. Math.133, 509–548 (1991)Google Scholar
  48. [Z] Zhang, S.: Ample Hermitian line bundles on arithmetic surfaces. (Preprint 1990)Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Henri Gillet
    • 1
  • Christophe Soulé
    • 2
  1. 1.Department of MathematicsUniversity of Illinois at ChicagoChicagoUSA
  2. 2.I.H.E.S. and C.N.R.S. MathématiquesBures-Sur-YvetteFrance

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