Duke Mathematical Journal

Capacity theory and arithmetic intersection theory

Ted Chinburg, Chi Fong Lau, and Robert Rumely

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Abstract

We show that the sectional capacity of an adelic subset of a projective variety over a number field is a quasi-canonical limit of arithmetic top self-intersection numbers, and we establish the functorial properties of extremal plurisubharmonic Green's functions. We also present a conjecture that the sectional capacity should be a top selfintersection number in an appropriate adelic arithmetic intersection theory.

Article information

Source
Duke Math. J., Volume 117, Number 2 (2003), 229-285.

Dates
First available in Project Euclid: 26 May 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1085598370

Digital Object Identifier
doi:10.1215/S0012-7094-03-11722-6

Mathematical Reviews number (MathSciNet)
MR1971294

Zentralblatt MATH identifier
1026.11056

Subjects
Primary: 11G35: Varieties over global fields [See also 14G25]
Secondary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30] 32U20: Capacity theory and generalizations 32U35: Pluricomplex Green functions

Citation

Chinburg, Ted; Lau, Chi Fong; Rumely, Robert. Capacity theory and arithmetic intersection theory. Duke Math. J. 117 (2003), no. 2, 229--285. doi:10.1215/S0012-7094-03-11722-6. https://projecteuclid.org/euclid.dmj/1085598370


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References

  • A. Abbes, Hauteurs et discrétude (d'après L. Szpiro, E. Ullmo et S. Zhang), Astérisque 245 (1997), 141--166., Séminaire Bourbaki 1996/97, exp. 825.
  • A. Abbes and T. Bouche, Théorème de Hilbert-Samuel ``arithmétique,'' Ann. Inst. Fourier (Grenoble) 45 (1995), 345--401.
  • P. Autissier, Points entiers sur les surfaces arithmétiques, J. Reine Angew. Math. 531 (2001), 201--235.
  • E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1--40.
  • V. G. Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Math. Surveys Monogr. 33, Amer. Math. Soc., Providence, 1990.
  • Y. Bilu, Limit distribution of small points on algebraic tori, Duke Math. J. 89 (1997), 465--476.
  • S. Bloch, H. Gillet, and C. Soulé, Non-Archimedean Arakelov theory, J. Algebraic Geom. 4 (1995), 427--485.
  • J.-B. Bost, Potential theory and Lefschetz theorems for arithmetic surfaces, Ann. Sci. École Norm. Sup. (4) 32 (1999), 241--312.
  • J.-B. Bost, H. Gillet, and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7 (1994), 903--1027.
  • T. Chinburg, Capacity theory on varieties, Compositio Math. 80 (1991), 75--84.
  • J.-P. Demailly, ``Monge-Ampère operators, Lelong numbers and intersection theory'' in Complex Analysis and Geometry, Univ. Ser. Math., Plenum, New York, 1993, 115--193.
  • R. Erné, On the degree of integral points of a projective space minus a horizontal hypersurface, J. Reine Angew. Math. 532 (2001), 151--177.
  • J. E. Fornaess and R. Narasimhan, The Levi problem on complex spaces with singularities, Math. Ann. 248 (1980), 47--72.
  • H. Gillet and C. Soulé, Amplitude Arithmétique, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 887--890.
  • --. --. --. --., An arithmetic Riemann-Roch theorem, Invent. Math. 110 (1992), 473--543.
  • R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
  • E. Kani, ``Potential theory on curves'' in Théorie des nombres (Quebec, 1987), de Gruyter, Berlin, 1989, 475--543.
  • M. Klimek, Pluripotential Theory, London Math. Soc. Monogr. (N.S.) 6, Oxford Univ. Press, New York, 1991.
  • P. Lelong, Plurisubharmonic Functions and Positive Differential Forms, trans. M. A. Dostal, Notes on Math. and Its Appl., Gordon and Breach, New York, 1969.
  • S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 449--474.
  • V. Maillot, Géométrie d'Arakelov des variétés toriques et fibrés en droites intégrables, Mém. Soc. Math. Fr. (N.S.) 80, Soc. Math. France, Marseille, 2000.
  • P. Mikkelson, Effective bounds for integral points on arithmetic surfaces, Ph.D. dissertation, Columbia University, New York, 1995.
  • J. S. Milne, Étale Cohomology, Princeton Math. Ser. 33, Princeton Univ. Press, Princeton, 1980.
  • L. Moret-Bailly, Groupes de Picard et problèmes de Skolem, I, II, Ann. Sci. École Norm. Sup. (4) 22 (1989), 161--179., 181--194.
  • --. --. --. --., ``Applications of local-global principles to arithmetic and geometry'' in Hilbert's Tenth Problem: Relations with Arithmetic and Algebraic Geometry (Ghent, Belgium, 1999), Contemp. Math. 270, Amer. Math. Soc., Providence, 2000, 169--186.
  • D. Mumford, Algebraic Geometry, I: Complex Projective Varieties, Grundlehren Math. Wiss. 221, Springer, Berlin, 1976.
  • R. Richberg, Stetige streng pseudokonvexe Funktionen, Math. Ann. 175 (1968), 257--286.
  • R. S. Rumely, Arithmetic over the ring of all algebraic integers, J. Reine Angew. Math. 368 (1986), 127--133.
  • --------, Capacity Theory on Algebraic Curves, Lecture Notes in Math. 1378, Springer, Berlin, 1989.
  • --. --. --. --., On the relation between Cantor's capacity and the sectional capacity, Duke Math. J. 70 (1993), 517--574.
  • --. --. --. --., ``An intersection pairing for curves, with analytic contributions from non-Archimedean places'' in Number Theory (Halifax, Nova Scotia, 1994), CMS Conf. Proc. 15, Amer. Math. Soc., Providence, 1995, 325--357.
  • R. Rumely and C. F. Lau, Arithmetic capacities on $\mathbbP^n$, Math. Z. 215 (1994), 533--560.
  • R. Rumely, C. F. Lau, and R. Varley, Existence of the Sectional Capacity, Mem. Amer. Math. Soc. 145 (2000), no. 690.
  • J. Siciak, On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc 105 (1962), 322--357.
  • --. --. --. --., Extremal plurisubharmonic functions in $\mathbbC^n$, Ann. Polon. Math. 39 (1981), 175--211.
  • L. Szpiro, E. Ullmo, and S. Zhang, Équirépartition des petits points, Invent. Math. 127 (1997), 337--347.
  • E. Ullmo, Positivité et discrétion des points algébriques des courbes, Ann. of Math. (2) 147 (1998), 167--179.
  • P. Vojta, Diophantine Approximations and Value Distribution Theory, Lecture Notes in Math. 1239, Springer, Berlin, 1987.
  • A. Werner, Non-Archimedean intersection indices on projective spaces and the Bruhat-Tits building for $\PGL$, Ann. Inst. Fourier (Grenoble) 51 (2001), 1483--1505.
  • A. Zériahi, Fonction de Green pluricomplexe à pôle à l'infini sur un espace de Stein parabolique et applications, Math. Scand. 69 (1991), 89--126.
  • S. Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. (2) 136 (1992), 569--587.
  • --. --. --. --., Admissible pairing on a curve, Invent. Math. 112 (1993), 171--193.
  • --. --. --. --., Positive line bundles on arithmetic varieties, J. Amer. Math. Soc 8 (1995), 187--221.
  • --. --. --. --., Small points and adelic metrics, J. Algebraic Geom. 4 (1995), 281--300.
  • --. --. --. --., ``Small points and Arakelov theory'' in Proceedings of the International Congress of Mathematicians (Berlin, 1998), Vol. II, Doc. Math. 1998, extra vol. II, 217--225.