Continuity of volumes on arithmetic varieties
Author:
Atsushi Moriwaki
Journal:
J. Algebraic Geom. 18 (2009), 407-457
DOI:
https://doi.org/10.1090/S1056-3911-08-00500-6
Published electronically:
May 13, 2008
MathSciNet review:
2496453
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Abstract |
References |
Additional Information
Abstract: We introduce the volume function for $C^{\infty }$-hermitian invertible sheaves on an arithmetic variety as an analogue of the geometric volume function. The main result of this paper is the continuity of the arithmetic volume function. As a consequence, we have the arithmetic Hilbert-Samuel formula for a nef $C^{\infty }$-hermitian invertible sheaf. We also give other applications, for example, a generalized Hodge index theorem, an arithmetic Bogomolov-Gieseker’s inequality, etc.
References
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References
- A. Abbes and T. Bouche, Théorème de Hilbert-Samuel “Arithmétique”, Ann. Inst. Fourier 45 (1995), 375-401. MR 1343555 (96e:14024)
- J.-M. Bismut and E. Vasserot, The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle, Comm. Math. Physics 125 (1989), 355-367. MR 1016875 (91c:58141)
- T. Bouche, Convergence de la metrique de Fubini-Study d’un fibre lineaire positif, Ann Inst. Fourier 40 (1990), 117-130. MR 1056777 (91d:32040)
- J. Bourgain and V.D. Milman, New volume ratio properties for convex symmetric bodies in $\mathbb {R}^n$, Invent. Math. 88 (1987), 319-340. MR 880954 (88f:52013)
- G. Faltings, Calculus an arithmetic surfaces, Ann. of Math. 119 (1984), 387-424. MR 740897 (86e:14009)
- H. Gillet and C. Soulé, On the number of lattice points in convex symmetric bodies and their dual, Israel J. Math. 74 (1991), 347-357. MR 1135244 (92k:11069)
- H. Gillet and C. Soulé, An arithmetic Riemann-Roch theorem, Invent. Math. 110 (1992), 473-543. MR 1189489 (94f:14019)
- P. M. Gruber and C.G. Lekkerkerker, Geometry of numbers, North-Holland Mathematical Library, Vol. 37, North-Holland. MR 893813 (88j:11034)
- H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, II, Ann. of Math. 79 (1964), 109–208. MR 0199184 (33:7333)
- P. Hriljac, Heights and Arakelov’s intersection theory, Amer. J. Math. 107 (1985), 23-38. MR 778087 (86c:14024)
- S. Kawaguchi, A. Moriwaki and K. Yamaki, Introduction to Arakelov theory, Proceedings of the Symposium on Algebraic Geometry in East Asia (2002), 1–74, World Scientific Publishing. MR 2030448 (2004k:14040)
- S. L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293–344 MR 0206009 (34:5834)
- R. Lazarsfeld, Positivity in Algebraic Geometry I, II, Springer-Verlag. MR 2095471 (2005k:14001a)
- A. Moriwaki, Inequality of Bogomolov-Gieseker type on arithmetic surfaces, Duke Math. J. 74 (1994), 713–761. MR 1277952 (95i:14025)
- A. Moriwaki, Arithmetic Bogomolov-Gieseker’s inequality, Amer. J. Math. 117 (1995), 1325–1347. MR 1350599 (96i:14022)
- A. Moriwaki, Arithmetic height functions over finitely generated fields, Invent. Math. 140 (2000), 101–142. MR 1779799 (2001g:11104)
- S. Takagi, Fujita’s approximation theorem in positive characteristics, J. Math. Kyoto Univ. 47 (2007), 179-202.
- G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Diff. Geometry 32 (1990), 99-130. MR 1064867 (91j:32031)
- X. Yuan, Big line bundles over arithmetic varieties, preprint.
- S. Zhang, Positive line bundles on arithmetic varieties, J. AMS 8 (1995), 187-221. MR 1254133 (95c:14020)
Additional Information
Atsushi Moriwaki
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502, Japan
Email:
moriwaki@math.kyoto-u.ac.jp
Received by editor(s):
January 22, 2007
Received by editor(s) in revised form:
September 14, 2007
Published electronically:
May 13, 2008
