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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 $$O_X$$.
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.