Abstract
In this paper we prove a refined version of the canonical key formula for projective abelian schemes in the sense of Moret-Bailly (cf. Astérisque 129, 1985), we also extend this discussion to the context of Arakelov geometry. Precisely, let \({\pi: A \to S}\) be a projective abelian scheme over a locally noetherian scheme S with unit section \({e: S \to A}\) and let L be a symmetric, rigidified, relatively ample line bundle on A. Denote by ω A the determinant of the sheaf of differentials of π and by d the rank of the locally free sheaf π* L. In this paper, we shall prove the following results: (i). there is an isomorphism
which is canonical in the sense that it can be chosen to be functorial, namely it is compatible with arbitrary base-change; (ii). if the generic fibre of S is separated and smooth, then there exist a positive integer m and canonical metrics on L and on ω A such that there exists an isometry
which is canonical in the sense of (i). Here the constant m only depends on g, d and is independent of L.
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Berline N., Getzler E., Vergne M.: Heat Kernels and the Dirac Operator, Grundlehren der Math., Wiss. 298. Springer, Berlin (1992)
Bismut J.-M., Koehler K.: Higher analytic torsion and anomaly formulas. J. Algebraic Geom. 1, 647–684 (1992)
Burgos Gil, J.I., Freixas i Montplet, G., Liţcanu, R.: Generalized holomorphic analytic torsion. J. Eur. Math. Soc. (JEMS) 16(3), 463–535 (2014)
Chai C.-L.: Siegel Moduli Schemes and Their Compactifications over \({\mathbb{C}}\), Arithmetic Geometry (Storrs, Conn., 1984), pp. 231–251. Springer, New York (1986)
Chai C.-L.: Compactification of Siegel Moduli Schemes, London Mathematical Society Lecture Note Series, vol. 107. Cambridge University Press, Cambridge (1985)
Faltings G., Chai C.-L.: Degeneration of Abelian Varieties, with an Appendix by David Mumford, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 22. Springer, Berlin (1990)
Freitag E., Kiehl R.: Étale Cohomology and the Weil Conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 13. Springer, Berlin (1988)
Fundamental algebraic geometry, Math. Surveys Monogr., 123. American Mathematical Society, Providence, RI (2005)
Gillet, H., Soulé, C.: Characteristic classes for algebraic vector bundles with hermitian metrics I, II, Ann. Math. 131, 163–203 and 205–238 (1990)
Gillet H., Soulé C.: An arithmetic Riemann–Roch theorem. Invent. Math. 110, 473–543 (1992)
Grothendieck A., Berthelot P., Illusie L.: SGA6, Théorie des Intersections et Théorème de Riemann–Roch, Lecture Notes in Maththematics 225. Springer, Berlin (1971)
Hartshorne R.: Algebraic Geometry, Graduate Texts in Mathematics 52. Springer, New York (1977)
Köck, B.: The Grothendieck–Riemann–Roch theorem for group scheme actions, Ann. Sci. Ecole Norm. Sup. 31, 4ème série, 415–458 (1998)
Ma X.: Formes de torsion analytique et familles de submersions I. Bull. Soc. Math. France 127, 541–562 (1999)
Maillot V., Rössler D.: On the determinant bundles of abelian schemes. Compositio Math. 144, 495–502 (2008)
Milne J.S.:Étale Cohomology, Princeton Mathematical Series, vol. 33. Princeton University Press, New Jersey (1980)
Moret-Bailly, L.: Pinceaux de variéeés abéliennes, Astérisque 129 (1985)
Moret-Bailly L.: Sur l’équation fonctionnelle de la fonction thêta de Riemann. Compositio Math. 75, 203–217 (1990)
Mumford D.: Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Oxford University Press, London (1970)
Mumford D., Fogarty J., Kirwan F.: Geometric Invariant Theory, Third Edition, Ergebnisse der Mathematik und Ihrer Grenzgebiete (2), vol. 34. Springer, Berlin (1994)
Polishchuk A.: Determinant bundles for abelian schemes. Compositio Math. 121, 221–245 (2000)
Rapoport M., Schappacher N., Schneider P.: Beilinson’s Conjectures on Special Values of L-functions, Perspect. Math. 4. Academic Press, Boston (1988)
Raynaud M.: Spécilisation du foncteur de Picard. Inst. Hautes Études Sci. Publ. Math. 38, 27–76 (1970)
Roessler D.: An Adams–Riemann–Roch theorem in Arakelov geometry. Duke Math. J. 96(1), 61–126 (1999)
Roessler D.: Lambda structure on arithmetic Grothendieck groups. Israel J. Math. 122, 279–304 (2001)
Soulé C.: Lectures on Arakelov Geometry, Cambridge Stud. Adv. Math. 33. Cambridge University Press, Cambridge (1991)
Vistoli A.: Grothendieck Topologies, Fibred Categories and Descent Theory, Fundamental Algebraic Geometry, 1–104, Math. Surveys Monogr., 123. American Mathematical Society, Providence, RI (2005)
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Research of the author was supported in part by NSFC (No. 11301352).
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Tang, S. Canonical key formula for projective abelian schemes. manuscripta math. 145, 255–284 (2014). https://doi.org/10.1007/s00229-014-0674-x
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DOI: https://doi.org/10.1007/s00229-014-0674-x