Kyoto Journal of Mathematics

Toward Dirichlet’s unit theorem on arithmetic varieties

Atsushi Moriwaki

Full-text: Open access

Abstract

In this paper, we would like to propose a fundamental question about a higher-dimensional analogue of Dirichlet’s unit theorem. We also give a partial answer to the question as an application of the arithmetic Hodge index theorem.

Article information

Source
Kyoto J. Math., Volume 53, Number 1 (2013), 197-259.

Dates
First available in Project Euclid: 25 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1364218046

Digital Object Identifier
doi:10.1215/21562261-1966116

Mathematical Reviews number (MathSciNet)
MR3049312

Zentralblatt MATH identifier
1270.14008

Subjects
Primary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]
Secondary: 11G50: Heights [See also 14G40, 37P30] 37P30: Height functions; Green functions; invariant measures [See also 11G50, 14G40]

Citation

Moriwaki, Atsushi. Toward Dirichlet’s unit theorem on arithmetic varieties. Kyoto J. Math. 53 (2013), no. 1, 197--259. doi:10.1215/21562261-1966116. https://projecteuclid.org/euclid.kjm/1364218046


Export citation

References

  • [1] Z. Blocki and S. Kolodziej, On regularization of plurisubharmonic functions on manifolds, Proc. Amer. Math. Soc. 135 (2007), 2089–2093.
  • [2] J.-B. Bost, Potential theory and Lefschetz theorems for arithmetic surfaces, Ann. Sci. École Norm. Sup. 32 (1999), 241–312.
  • [3] J. I. Burgos Gil, A. Moriwaki, P. Philippon, and M. Sombra, Arithmetic positivity on toric varieties, preprint, arXiv:1210.7692v1 [math.AG]
  • [4] A. Chambert-Loir, Arakelov geometry: Heights and Bogomolov conjecture, preprint, 2009.
  • [5] H. Chen, Positive degree and arithmetic bigness, preprint, arXiv:0803.2583v3 [math.AG]
  • [6] A. J. de Jong, Smoothness, semi-stability, and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93.
  • [7] J.-P. Demailly, Complex analytic and differential geometry, preprint, 2007, http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf
  • [8] G. Faltings, Calculus on arithmetic surfaces, Ann. of Math. 119 (1984), 387–424.
  • [9] M. Hindry and J. H. Silverman, Diophantine Geometry: an Introduction, Grad. Texts in Math. 201, Springer, New York, 2000.
  • [10] P. Hriljac, Height and Arakerov’s intersection theory, Amer. J. Math. 107 (1985), 23–38.
  • [11] J. Jost, Postmodern Analysis, 3rd ed., Universitext, Springer, Berlin, 2005.
  • [12] J. Lipman, Desingularization of two-dimensional schemes, Ann. of Math. (2) 107 (1978), 151–207.
  • [13] V. Maillot, Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables, Mém. Soc. Math Fr. (N.S.) 83, Soc. Math. France, Montrouge, 2000.
  • [14] J. Milnor, Morse Theory, based on lecture notes by M. Spivak and R. Wells, Ann. of Math. Stud. 51, Princeton Univ. Press, Princeton, 1963.
  • [15] A. Moriwaki, Arithmetic Bogomolov-Gieseker’s inequality, Amer. J. Math. 117 (1995), 1325–1347.
  • [16] A. Moriwaki, Hodge index theorem for arithmetic cycles of codimension one, Math. Res. Lett. 3, 173–183 (1996).
  • [17] A. Moriwaki, Continuity of volumes on arithmetic varieties, J. Algebraic Geom. 18 (2009), 407–457.
  • [18] A. Moriwaki, Continuous extension of arithmetic volumes, Int. Math. Res. Not. IMRN (2009), no. 19, 3598–3638.
  • [19] A. Moriwaki, Estimation of arithmetic linear series, Kyoto J. Math. 50 (2010), 685–725.
  • [20] A. Moriwaki, Big arithmetic divisors on the projective spaces over $\mathbb{Z}$, Kyoto J. Math. 51 (2011), 503–534.
  • [21] A. Moriwaki, Zariski decompositions on arithmetic surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), 799–898.
  • [22] A. Moriwaki, Numerical characterization of nef arithmetic divisors on arithmetic surfaces, preprint, arXiv:1201.6124v4 [math.AG]
  • [23] L. Szpiro, “Degrés, intersections, hauteurs,” in Seminar on Arithmetic Bundles: The Mordell Conjecture (Paris, 1983/1984), Astérisque 127, Soc. Math. France, Montrouge, 1985, 11–28.
  • [24] S. Zhang, Positive line bundles on arithmetic varieties, J. Amer. Math. Soc. 8 (1995), 187–221.
  • [25] S. Zhang, Small points and adelic metrics, J. Algebraic Geom. 4 (1995), 281–300.