×

zbMATH — the first resource for mathematics

Continuity of volumes on arithmetic varieties. (English) Zbl 1167.14014
The notion of volume function is important in algebraic geometry. It is a birational invariant which measures the asymptotic behaviour of the sectional algebra of a line bundle on a projective variety. In the article under review, the author has studied the analogue of the volume function in Arakelov geometry, notably the continuity of the arithmetic volume function. Recall that the arithmetic volume function of a \(C^\infty\) Hermitian line bundle \(\overline L\) on an arithmetic variety \(X\) of total dimension \(d\) is defined as \[ \widehat{\mathrm{vol}}(\overline L):=\limsup_{m\rightarrow+\infty}\frac{\widehat{h}^0(X, m\overline L)}{m^d/d!}, \] where \(\widehat{h}^0(X,m\overline L)\) is defined as \(\log\#\{s\in H^0(X,L^{\otimes m})\,:\, \|s\|_{\sup}\leqslant 1\}\). By the equality \(\widehat{\mathrm{vol}}(m\overline L)=m^{d}\widehat{\mathrm{vol}}(\overline L)\), one can extend the definition domain of this function to \(\widehat{\mathrm{Pic}}(X)\otimes_{\mathbb Z}\mathbb Q\), where \(\widehat{\mathrm{Pic}}(X)\) denotes the group of smooth Hermitian line bundles on \(X\), whose group law is written additively. Namely, for all \(\overline L,\overline A_1,\cdots,\overline A_n\) in \(\widehat{\mathrm{Pic}}(X)\), there exists a positive constant \(C\) such that \[ \big|\widehat{\mathrm{vol}}(\overline L+\epsilon_1\overline A_1+\cdots+\epsilon_n\overline A_n)-\widehat{\mathrm{vol}}(\overline L)\big|\leqslant C\big(|\epsilon_1|+\cdots+|\epsilon_n|\big). \] The proof of this result uses an estimation of distorsion functions due to T. Bouche [Ann. Inst. Fourier 40, No. 1, 117–130 (1990; Zbl 0685.32015) and G. Tian [J. Differ. Geom. 32, No. 1, 99–130 (1990; Zbl 0706.53036)].
As an application, the author generalizes the arithmetic Hilbert-Samuel function to the nef case, in the following sense: assume that \(\overline L\) and \(\overline N\) are in \(\widehat{\mathrm{Pic}}(X)\) with \(\overline L\) nef, then \[ \widehat{h}^0(X,m\overline L+\overline N)=\frac{\widehat{c}_1(\overline L)^d}{d!}m^d+o(m^d)\qquad(m\rightarrow\infty). \] Consequently, a nef \(C^\infty\) Hermitian line bundle \(\overline L\) is big (i.e. \(\widehat{\mathrm{vol}}(\overline L)>0\)) if and only if \(\widehat{c}_1(\overline L)^d>0\). Similarly, for \(C^\infty\) Hermitian line bundle \(\overline L\) with nef generic fibre and semi-positive metric, if \(\overline L\) has moderate growth of positive even cohomologies, then the inequality \(\widehat{\mathrm{vol}}(\overline L)\geqslant \widehat{c}_1(\overline L)^d\) holds. This result generalizes the arithmetic Hodge index theorem due to G. Faltings [Ann. Math. (2) 119, 387–424 (1984; Zbl 0559.14005)] and P. Hriljac [Am. J. Math. 107, 23–38 (1985; Zbl 0593.14004)] and the arithmetic Bogomolov-Gieseker’s inequality.
Reviewer: Huayi Chen (Paris)

MSC:
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] 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). · Zbl 0818.14011
[2] 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. · Zbl 0687.32023
[3] 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). · Zbl 0685.32015
[4] J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in \?\(^{n}\), Invent. Math. 88 (1987), no. 2, 319 – 340. · Zbl 0617.52006
[5] Gerd Faltings, Calculus on arithmetic surfaces, Ann. of Math. (2) 119 (1984), no. 2, 387 – 424. · Zbl 0559.14005
[6] 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. · Zbl 0752.52008
[7] Henri Gillet and Christophe Soulé, An arithmetic Riemann-Roch theorem, Invent. Math. 110 (1992), no. 3, 473 – 543. · Zbl 0777.14008
[8] P. M. Gruber and C. G. Lekkerkerker, Geometry of numbers, 2nd ed., North-Holland Mathematical Library, vol. 37, North-Holland Publishing Co., Amsterdam, 1987. · Zbl 0611.10017
[9] 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. · Zbl 0122.38603
[10] Paul Hriljac, Heights and Arakelov’s intersection theory, Amer. J. Math. 107 (1985), no. 1, 23 – 38. · Zbl 0593.14004
[11] Shu Kawaguchi, Atsushi Moriwaki, and Kazuhiko Yamaki, Introduction to Arakelov geometry, Algebraic geometry in East Asia (Kyoto, 2001) World Sci. Publ., River Edge, NJ, 2002, pp. 1 – 74. · Zbl 1051.14028
[12] Steven L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293 – 344. · Zbl 0146.17001
[13] 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. Robert Lazarsfeld, Positivity in algebraic geometry. II, 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. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals.
[14] Atsushi Moriwaki, Inequality of Bogomolov-Gieseker type on arithmetic surfaces, Duke Math. J. 74 (1994), no. 3, 713 – 761. · Zbl 0854.14012
[15] Atsushi Moriwaki, Arithmetic Bogomolov-Gieseker’s inequality, Amer. J. Math. 117 (1995), no. 5, 1325 – 1347. · Zbl 0854.14013
[16] Atsushi Moriwaki, Arithmetic height functions over finitely generated fields, Invent. Math. 140 (2000), no. 1, 101 – 142. · Zbl 1007.11042
[17] S. Takagi, Fujita’s approximation theorem in positive characteristics, J. Math. Kyoto Univ. 47 (2007), 179-202. · Zbl 1136.14004
[18] Gang Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99 – 130. · Zbl 0706.53036
[19] X. Yuan, Big line bundles over arithmetic varieties, preprint. · Zbl 1146.14016
[20] Shouwu Zhang, Positive line bundles on arithmetic varieties, J. Amer. Math. Soc. 8 (1995), no. 1, 187 – 221. · Zbl 0861.14018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.