Equidistribution of small subvarieties of an Abelian variety.

*(English)*Zbl 1131.11035Let \(A\) be an Abelian variety over a number field \(K\). Using the choice of an embedding of \(\overline K\) into \(\mathbb{C}\), the set \(A(\overline K)\) may then be viewed as a subset of \(A(\mathbb{C})\), and any symmetric ample line bundle \(L\) on \(A\) can be equipped with a so-called cubical metric, i.e., a smooth Hermitean metric whose curvature form is invariant under translations in \(A\) [cf. L. Moret-Bailly, Exp. IV, Astérisque 127, 29–87 (1985; Zbl 0588.14028)]. The resulting metrized line bundle gives rise to a canonical height function \(\widehat h_L\) on the set of closed subvarieties of \(A\), which actually does not depend on the choice of a cubical metric on \(L\) [cf. S. Zhang, J. Algebr. Geom. 4, 281–300 (1995; Zbl 0861.14019)] and coincides with the classical Néron-Tate canonical height with respect to \(L\) on closed points \(x\in A(\overline K)\). Finally, a small strict sequence of closed subvarieties of \(A\) is a sequence \((X_n)_{n\in\mathbb{N}}\) of closed subvatieties such that \(\lim_{n\to\infty}\,\widehat h(X_n)= 0\) and such that no proper closed torsion subvariety of \(A\) contains an infinite subsequence of \((X_n)_{n\in\mathbb{N}}\).

With respect to this general set-up, the authors of the paper under review provide a detailed proof of the following equidistribution result:

Theorem 1.1. Let \(A\) be an Abelian variety over a number field \(K\), let \(L\) be a symmetric ample line bundle on \(A\), and let \((X_n)_{n\in\mathbb{N}}\) be a small strict sequence of closed subvarieties of \(A\). Then for each complex embedding of \(\overline K\) into \(\mathbb{C}\) and for every real-valued function \(f\) on \(A(\mathbb{C})\) the sequence of integrals \(\int_{A(\mathbb{C})} f\mu_n\) converges to the integral \(\int_{A(\mathbb{C})}f\mu\) as \(n\to\infty\), where setting \(d_n= \dim X_n\) and \(g= \dim A\) the occurring measures are \[ \mu_n= (c_1(L_{X_n})^{-d_n}\cdot c_1(L)^{d_n}\cdot \delta_{X_n}\quad \text{and}\quad \mu= c_1(\overline L)^g\cdot c_1(L)^{-g}. \] This amounts to saying that the sequence \((\mu_n)_{n\in\mathbb{N}}\) of measures weakly converges to the Haar measure \(\mu\) of total mass 1 on \(A(\mathbb{C})\).

This result generalizes the corresponding equidistribution theorem for small strict sequences of points in an Abelian variety \(A\) over \(K\), which was proven by L. Szpiro, E. Ullmo and S. Zhang [Invent. Math. 127, No. 2, 337–347 (1997; Zbl 0991.11035)].

In fact, the authors’ proof of their higher-dimensional equidistribution theorem is partially based on the foregoing Szpiro-Ullmo-Zhang approach. Namely, they first show how to approximate the height of each subvariety \(X_n\) by Néron-Tate heights of points on \(X_n\) thereby using S.-W. Zhang’s theorem [Ann. Math. (2) 147, No. 1, 159–165 (1998; Zbl 0991.11034)], and apply then the method of proof of Szpiro-Ullmo-Zhang to a particular subsequence of such points.

With respect to this general set-up, the authors of the paper under review provide a detailed proof of the following equidistribution result:

Theorem 1.1. Let \(A\) be an Abelian variety over a number field \(K\), let \(L\) be a symmetric ample line bundle on \(A\), and let \((X_n)_{n\in\mathbb{N}}\) be a small strict sequence of closed subvarieties of \(A\). Then for each complex embedding of \(\overline K\) into \(\mathbb{C}\) and for every real-valued function \(f\) on \(A(\mathbb{C})\) the sequence of integrals \(\int_{A(\mathbb{C})} f\mu_n\) converges to the integral \(\int_{A(\mathbb{C})}f\mu\) as \(n\to\infty\), where setting \(d_n= \dim X_n\) and \(g= \dim A\) the occurring measures are \[ \mu_n= (c_1(L_{X_n})^{-d_n}\cdot c_1(L)^{d_n}\cdot \delta_{X_n}\quad \text{and}\quad \mu= c_1(\overline L)^g\cdot c_1(L)^{-g}. \] This amounts to saying that the sequence \((\mu_n)_{n\in\mathbb{N}}\) of measures weakly converges to the Haar measure \(\mu\) of total mass 1 on \(A(\mathbb{C})\).

This result generalizes the corresponding equidistribution theorem for small strict sequences of points in an Abelian variety \(A\) over \(K\), which was proven by L. Szpiro, E. Ullmo and S. Zhang [Invent. Math. 127, No. 2, 337–347 (1997; Zbl 0991.11035)].

In fact, the authors’ proof of their higher-dimensional equidistribution theorem is partially based on the foregoing Szpiro-Ullmo-Zhang approach. Namely, they first show how to approximate the height of each subvariety \(X_n\) by Néron-Tate heights of points on \(X_n\) thereby using S.-W. Zhang’s theorem [Ann. Math. (2) 147, No. 1, 159–165 (1998; Zbl 0991.11034)], and apply then the method of proof of Szpiro-Ullmo-Zhang to a particular subsequence of such points.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

11G10 | Abelian varieties of dimension \(> 1\) |

11G35 | Varieties over global fields |

14K15 | Arithmetic ground fields for abelian varieties |

14K12 | Subvarieties of abelian varieties |

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |

28C10 | Set functions and measures on topological groups or semigroups, Haar measures, invariant measures |