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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.

MSC:
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
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References:
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