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Points of small height on semiabelian varieties. (English) Zbl 1487.14063

Let \(K\) be a number field, and \(X\) a projective variety over \(K\). For a place \(\nu\) of \(K\), denote by \(\mathbb{C}_{\nu} := \widehat{\overline{K}_{\nu}}\) and \(X_{\mathbb{C}_{\nu}}^{\operatorname{an}}\) the corresponding analytic space. A “generic” sequence of closed points \(\{x_i\}\) of \(X\) is a sequence such that no infinite subsequence is contained in a proper closed subvariety of \(X\).
Let \(\overline{L}\) be a line bundle on \(X\) endowed with a set of metrics; this defines a height function on closed points of \(X\), which we denote by \(h_{\overline{L}}\), which we assume for now to be positive. A “small” sequence of closed points \(\{x_i\}\) of \(X\) is a sequence such that \(h_{\overline{L}}(x_i)\to 0\).
Then the equidistribution conjecture of small points can be formulated in the following way.
Conjecture 1 (Equidistribution conjecture): Let \(\{x_i\}\) be a generic small sequence of closed points of \(X\). Then, for every place \(\nu\) of \(K\), the measures \[ \frac{1}{\#O_{\nu}(x_i)}\sum_{y \in O_{\nu}(x_i)}\delta_y \; \text{ converge weakly to } \frac{1}{\operatorname{deg}_L(X)}c_1(\overline{L}_{\nu})^{\wedge \operatorname{dim}(X)}, \] where \(O_{\nu}(x_i) = \left(x_i \otimes_K\mathbb{C}_{\nu}\right)^{\operatorname{an}}\) is the analytic \(0\)-cycle of \(X_{\mathbb{C}_{\nu}}^{\operatorname{an}}\) associated to \(x_i\), \(\delta_y\) is the Dirac measure supported at \(y\), and \(c_1\left(\overline{L}_{\nu}\right)^{\wedge \operatorname{dim}(X)}\) is a measure associated to \(\overline{L}_{\nu}\).
When \(X\) is an algebraic group and \(L\) is a line bundle on \(X\), there is a canonical way of associating a metric to \(L\) (hence defining a height function). For instance, these height functions uniquely identify torsion points of \(X\left(\overline{K}\right)\) with closed points of zero height. In this context, one can state the following conjecture.
Conjecture 2 (Bogomolov conjecture): Let \(Y\) be a geometrically irreducible algebraic subvariety of \(X\), which is not an irreducible component of an algebraic subgroup of \(X\). Then there exists an \(\varepsilon >0\) such that the set \[ \left\{y \in Y\left(\overline{K}\right) \; | \; h_{\overline{L}}(y) \leq \varepsilon \right\} \] is not Zariski dense in \(X\).
The equidistribution conjecture was proven for abelian varieties by L. Szpiro et al. [Invent. Math. 127, No. 2, 337–347 (1997; Zbl 0991.11035)] where they initiated a method relaying on the arithmetic Hilbert–Samuel theorem. A more general equidistribution theorem was then proven by Yuan who extends their principle (see [X. Yuan, Invent. Math. 173, No. 3, 603–649 (2008; Zbl 1146.14016)]). These techniques are however not useful in the case of semiabelian varieties, since they relay on generic sequences of points whose height converges towards the height of the ambient variety, which in the case of semiabelian varieties is negative unless it is almost split. On the other hand, the Bogomolov conjecture was settled for abelian varieties and algebraic tori by S. Zhang [J. Amer. Math. Soc. 8, No. 1, 187–221 (1995; Zbl 0861.14018); Ann. of Math. (2) 147, No. 1, 159–165 (1998; Zbl 0991.11034)]. Regarding semiabelian varieties, both conjectures were only known whenever the abelian variety is “almost split” due to the work of A. Chambert-Loir [Ann. Sci. École Norm. Sup. (4) 33, No. 6, 789–821 (2000; Zbl 1018.11034)], where his main obstruction is the negativity of the hieght of semiabelian varieties in the non-split case. The Bogomolov conjecture was proven by S. David and P. Philippon [C. R. Acad. Sci. Paris Sér. I Math 331, 387–592 (2000; Zbl 0972.11059)] using a different approach.
In the present article, the author proves both statements in the case of general semiabelian varieties using an asymptotoc adaption of the techniques initiated by Szpiro, Ullmo and Zhang in [loc. cit.] avoiding the main obstructions occurring in the work of Chambert-Loir.

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
11G10 Abelian varieties of dimension \(> 1\)
14G05 Rational points
14K15 Arithmetic ground fields for abelian varieties
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