Boundedness of the successive minima on arithmetic varieties
Author:
Hideaki Ikoma
Journal:
J. Algebraic Geom. 22 (2013), 249-302
DOI:
https://doi.org/10.1090/S1056-3911-2012-00600-6
Published electronically:
July 10, 2012
MathSciNet review:
3019450
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Abstract |
References |
Additional Information
Abstract: In this paper, we study the asymptotic behavior of the successive minima associated with high powers of a Hermitian invertible sheaf on an arithmetic variety. As a consequence, we prove that the arithmetic $\hat {\chi }$-volume function, which is introduced by Yuan, is homogeneous, birationally invariant, and continuous on the arithmetic Picard group. We also obtain the arithmetic Hilbert-Samuel formula for vertically nef Hermitian invertible sheaves.
References
- A. Abbes and T. Bouche, Théorème de Hilbert-Samuel “arithmétique”, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 375–401 (French, with English and French summaries). MR 1343555
- Jean-Michel Bismut and Éric Vasserot, The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle, Comm. Math. Phys. 125 (1989), no. 2, 355–367. MR 1016875
- Thierry Bouche, Convergence de la métrique de Fubini-Study d’un fibré linéaire positif, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 1, 117–130 (French, with English summary). MR 1056777
- S. Boucksom and H. Chen, Okounkov bodies of filtered linear series, preprint (arXiv:0911.2923), 2009, to appear in Compos. Math.
- N. Bourbaki, Éléments de mathématique. Topologie générale. Chapitre 5 à 10, Springer-Verlag (2007).
- Huayi Chen, Arithmetic Fujita approximation, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), no. 4, 555–578 (English, with English and French summaries). MR 2722508, DOI https://doi.org/10.24033/asens.2127
- Huayi Chen, Convergence des polygones de Harder-Narasimhan, Mém. Soc. Math. Fr. (N.S.) 120 (2010), 116 (French, with English and French summaries). MR 2768967
- H. Gillet and C. Soulé, On the number of lattice points in convex symmetric bodies and their duals, Israel J. Math. 74 (1991), no. 2-3, 347–357. MR 1135244, DOI https://doi.org/10.1007/BF02775796
- Henri Gillet and Christophe Soulé, An arithmetic Riemann-Roch theorem, Invent. Math. 110 (1992), no. 3, 473–543. MR 1189489, DOI https://doi.org/10.1007/BF01231343
- P. M. Gruber and C. G. Lekkerkerker, Geometry of numbers, 2nd ed., North-Holland Mathematical Library, vol. 37, North-Holland Publishing Co., Amsterdam, 1987. MR 893813
- Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI https://doi.org/10.2307/1970547
- Jean-Pierre Jouanolou, Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42, Birkhäuser Boston, Inc., Boston, MA, 1983 (French). MR 725671
- Tommaso de Fernex, Alex Küronya, and Robert Lazarsfeld, Higher cohomology of divisors on a projective variety, Math. Ann. 337 (2007), no. 2, 443–455. MR 2262793, DOI https://doi.org/10.1007/s00208-006-0044-4
- Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471
- Atsushi Moriwaki, Continuity of volumes on arithmetic varieties, J. Algebraic Geom. 18 (2009), no. 3, 407–457. MR 2496453, DOI https://doi.org/10.1090/S1056-3911-08-00500-6
- Atsushi Moriwaki, Continuous extension of arithmetic volumes, Int. Math. Res. Not. IMRN 19 (2009), 3598–3638. MR 2539186, DOI https://doi.org/10.1093/imrn/rnp068
- A. Moriwaki, Zariski decompositions on arithmetic surfaces, preprint (arXiv:0911. 2951), 2009.
- A. Moriwaki, Big arithmetic divisors on projective spaces over $\mathbb {Z}$, preprint (arXiv:1003. 1782), 2010.
- Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR 2104208
- Robert Rumely, Chi Fong Lau, and Robert Varley, Existence of the sectional capacity, Mem. Amer. Math. Soc. 145 (2000), no. 690, viii+130. MR 1677934, DOI https://doi.org/10.1090/memo/0690
- Gang Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99–130. MR 1064867
- Xinyi Yuan, Big line bundles over arithmetic varieties, Invent. Math. 173 (2008), no. 3, 603–649. MR 2425137, DOI https://doi.org/10.1007/s00222-008-0127-9
- Xinyi Yuan, On volumes of arithmetic line bundles, Compos. Math. 145 (2009), no. 6, 1447–1464. MR 2575090, DOI https://doi.org/10.1112/S0010437X0900428X
- X. Yuan, On volumes of arithmetic line bundles II, preprint (arXiv:0909.3680), 2009.
- Shouwu Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. (2) 136 (1992), no. 3, 569–587. MR 1189866, DOI https://doi.org/10.2307/2946601
- Shouwu Zhang, Positive line bundles on arithmetic varieties, J. Amer. Math. Soc. 8 (1995), no. 1, 187–221. MR 1254133, DOI https://doi.org/10.1090/S0894-0347-1995-1254133-7
References
- A. Abbes and T. Bouche, Théorème de Hilbert-Samuel \og arithmetique\fg, Ann. Inst. Fourier (Grenoble) 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. Phys. 125 (1989), 355–367. MR 1016875 (91c:58141)
- T. Bouche, Convergence de la métrique de Fubini-Study d’un fibré linéaire positif, Ann. Inst. Fourier (Grenoble) 40 (1990), 117–130. MR 1056777 (91d:32040)
- S. Boucksom and H. Chen, Okounkov bodies of filtered linear series, preprint (arXiv:0911.2923), 2009, to appear in Compos. Math.
- N. Bourbaki, Éléments de mathématique. Topologie générale. Chapitre 5 à 10, Springer-Verlag (2007).
- H. Chen, Arithmetic Fujita approximation, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), 555–578. MR 2722508
- H. Chen, Convergence des polygones de Harder-Narashimhan, Mém. Soc. Math. Fr. (N.S.) 120 (2010). MR 2768967
- H. Gillet and C. Soulé, On the number of lattice points in convex symmetric bodies and their duals, Isr. 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, 2nd ed. North-Holland, Amsterdam (1987). MR 893813 (88j:11034)
- H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. 205–326. MR 0199184 (33:7333)
- J.-P. Jouanolou, Théorèmes de Bertini et applications, Birkhäuser (1983). MR 725671 (86b:13007)
- T. de Fernex, A. Küronya, and R. Lazarsfeld, Higher cohomology of divisors on a projective variety, Math. Ann. 337 (2007), 443–455. MR 2262793 (2008c:14010)
- R. Lazarsfeld, Positivity in algebraic geometry I, II, Springer-Verlag (2004). MR 2095471 (2005k:14001a)
- A. Moriwaki, Continuity of volumes on arithmetic varieties, J. Algebraic Geom. 18 (2009), 407–457. MR 2496453 (2011a:14053)
- A. Moriwaki, Continuous extension of arithmetic volumes, Int. Math. Res. Not. IMRN (2009), 3598–3638. MR 2539186 (2010j:14050)
- A. Moriwaki, Zariski decompositions on arithmetic surfaces, preprint (arXiv:0911. 2951), 2009.
- A. Moriwaki, Big arithmetic divisors on projective spaces over $\mathbb {Z}$, preprint (arXiv:1003. 1782), 2010.
- N. Nakayama, Zariski-decomposition and abundance, MSJ Mem. 14 (2004). MR 2104208 (2005h:14015)
- R. Rumely, C. F. Lau, and R. Varley, Existence of the sectional capacity, Mem. Amer. Math. Soc. 145 (2000). MR 1677934 (2000j:14039)
- G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), 99–130. MR 1064867 (91j:32031)
- X. Yuan, Big line bundles over arithmetic varieties, Invent. Math. 173 (2008), no. 3, 603–649. MR 2425137 (2010b:14049)
- X. Yuan, On volumes of arithmetic line bundles, Compos. Math. 145 (2009), 1447–1464. MR 2575090 (2011a:14054)
- X. Yuan, On volumes of arithmetic line bundles II, preprint (arXiv:0909.3680), 2009.
- S. Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. (2) 136 (1992), 569–587. MR 1189866 (93j:14024)
- S. Zhang, Positive line bundles on arithmetic varieties, J. Amer. Math. Soc. 8 (1995), 187–221. MR 1254133 (95c:14020)
Additional Information
Hideaki Ikoma
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502, Japan
Email:
ikoma@math.kyoto-u.ac.jp
Received by editor(s):
May 27, 2010
Received by editor(s) in revised form:
May 4, 2011
Published electronically:
July 10, 2012
Dedicated:
Dedicated to the memory of Professor Masaki Maruyama.