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Equidistribution over function fields. (English) Zbl 1189.14030
In this paper the author transfers the equidistribution results of Yuan in [X. Yuan, “Big line bundles on arithmetic varieties”, arXiv: math.NT/0612424 (2006)] to the context of function fields, make use of the generic curve to reduce the work to function fields of curves and manage to prove a variational version of the fundamental inequality to get an equidistribution theorem over function fields as inspired by the work Szpiro-Ullmo-Zhang in [L. Szpiro, E. Ullmo and S. Zhang, “Equirépartition despetits points”, Invent. Math. 127, No. 2, 337–347 (1997; Zbl 0991.11035)].
The skeleton of the proof is as follows:
a) Lemma 5.3 and corollary 5.4 will provide with nef metrics on semipositive admissible divisors.
b) An application of Siu’s theorem will give big line bundles and in particular a section \(s \in H^0({(L_r \otimes N^{\varepsilon})}^m)\) for \(-c<\varepsilon <c\) and some \(c>0\).
c) The section \(s\) as above will allow us to have \(h_{L_r \otimes N^{\varepsilon}}(P) \geq 0\) by theorem 3.5 e).
d) As a consequence the following version of the fundamental inequality is obtained: \[ \dfrac{h_{(L,\|.\| \otimes \|.\|_f^{\varepsilon})}(X)}{(d+1)\deg_L(X)} \leq e_1(X,_(L,\|.\| \otimes \|.\|_f^{\varepsilon})) + S \varepsilon^2, \] for some constant \(S\), \(\varepsilon \in (-c,c)\) for some \(c>0\), \(e_1(X,_(L,\|.\| \otimes \|.\|_f^{\varepsilon}))\) representing the essential minimum of definition 5.1 and \(\|.\|_f\) a formal \(M_C\) metric on \(O_X\).
e) The equidistribution theorem reads:
Theorem: Let \(L\) be a big semiample line bundle on the irreducible \(d\)-dimensional projective variety \(X\) over the function field \(K\). We endow \(L\) with a semipositive admissible metric \(\|.\|\). We assume that \((P_m)_{m \in I}\) is a generic and small net in \(X(\bar{K})\) with \[ \lim_m h_{(L,\|.\|)}(P_m) = \dfrac{1}{(d+1) deg_L(X)}h_{(L,\|.\|)}(X). \] For a place \(v\) of \(K\), we have the following weak limit of regular probability measures on \(X_v^{an}\): \[ \dfrac{1}{|O(P_m)|} \sum_{P_m^{\sigma} \in O(P_m)} \delta_{P_m^{\sigma}} \rightarrow \dfrac{1}{deg_L(X)} c_1(L,\|.\|_v)^d, \] where \(c_1(L,\|.\|_v)^d\) represents the Chambert-Loir measure on \(X_v^{an}\) .
In the case that \(X\) is a closed subvariety of an abelian variety, the author describes the equidistribution measure in terms of convex geometry.

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14T05 Tropical geometry (MSC2010)
11G50 Heights
37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
14C40 Riemann-Roch theorems
14G25 Global ground fields in algebraic geometry
37P55 Arithmetic dynamics on general algebraic varieties
Full Text: DOI arXiv
[1] Autissier P. (2006) Équidistribution de sous-variétés de petite hauteur. J. Théor. Nombres Bordx. 18(1): 1–12 · Zbl 1228.11087
[2] Baker M., Ih S.-I. (2004) Equidistribution of small subvarieties of an abelian variety. N. Y. J. Math. 10: 279–289 · Zbl 1131.11035
[3] Berkovich V.G. (1990) Spectral theory and analytic geometry over non-archimedean fields. Mathematical Surveys and Monographs, 33. AMS, Providence · Zbl 0715.14013
[4] Bilu Y. (1997) Limit distribution of small points on algebraic tori. Duke Math. J. 89(3): 465–476 · Zbl 0918.11035 · doi:10.1215/S0012-7094-97-08921-3
[5] Bombieri E., Gubler W. (2006) Heights in Diophantine Geometry. Cambridge University Press, Cambridge · Zbl 1115.11034
[6] Bosch S., Güntzer U., Remmert R. (1984) Non-Archimedean analysis. A systematic approach to rigid analytic geometry. Grundl. Math. Wiss., vol. 261. Springer, Berlin · Zbl 0539.14017
[7] Bosch S., Lütkebohmert W. (1993) Formal and rigid geometry. I: Rigid spaces. Math. Ann. 295(2): 291–317 · Zbl 0808.14017 · doi:10.1007/BF01444889
[8] Chambert-Loir A. (2006) Mesure et équidistribution sur les espaces de Berkovich. J. Reine Angew. Math. 595: 215–235 · Zbl 1112.14022 · doi:10.1515/CRELLE.2006.049
[9] Faber, X.W.C.: Equidistribution of dynamically small subvarieties over the function field of a curve, arXiv:math.NT:0801.4811v2 (preprint)
[10] Gubler W. (1998) Local heights of subvarieties over non-archimedean fields. J. Reine Angew. Math. 498: 61–113 · Zbl 0906.14013 · doi:10.1515/crll.1998.054
[11] Gubler W. (2003) Local and canonical heights of subvarieties. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (Ser. V) 2(4): 711–760 · Zbl 1170.14303
[12] Gubler W. (2007) Tropical varieties for non-archimedean analytic spaces. Invent. Math. 169(2): 321–376 · Zbl 1153.14036 · doi:10.1007/s00222-007-0048-z
[13] Gubler W. (2007) The Bogomolov conjecture for totally degenerate abelian varieties. Invent. Math. 169(2): 377–400 · Zbl 1153.14029 · doi:10.1007/s00222-007-0049-y
[14] Gubler, W.: Non-archimedean canonical measures on abelian varieties, available at arXiv:math.NT:0801.4503v1 (preprint)
[15] Lang S. (1958) Introduction to Algebraic Geometry. Interscience Publishers, New York · Zbl 0095.15301
[16] Lazarsfeld R. (2004) Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Bd. 48. Springer, Berlin · Zbl 1093.14501
[17] Szpiro L., Ullmo E., Zhang S. (1997) Equirépartition des petits points. Invent. Math. 127: 337–347 · Zbl 0991.11035 · doi:10.1007/s002220050123
[18] Ullmo E. (1998) Positivité et discrétion des points algébriques des courbes. Ann. Math. (2) 147(1): 167–179 · Zbl 0934.14013 · doi:10.2307/120987
[19] Ullrich P. (1995) The direct image theorem in formal and rigid geometry. Math. Ann. 301(1): 69–104 · Zbl 0821.32029 · doi:10.1007/BF01446620
[20] Yuan, X.: Positive line bundles over arithmetic varieties, available at arXiv:math. NT:0612424v1 (preprint)
[21] Zhang S. (1998) Equidistribution of small points on abelian varieties. Ann. Math. (2) 147(1): 159–165 · Zbl 0991.11034 · doi:10.2307/120986
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