The Bounded Height Conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian $\overline{\mathbb{Q}}$-variety $G$ there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in $G$. This conjecture has been shown by Habegger in the case where $G$ is either a multiplicative torus or an abelian variety. However, there are new obstructions to his approach if $G$ is a general semiabelian variety. In particular, the lack of Poincar\'e reducibility means that quotients of a given semiabelian variety are intricate to describe. To overcome this, we study directly certain families of line bundles on $G$. This allows us to demonstrate the conjecture for general semiabelian varieties.

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半阿贝尔变体的有界高度猜想

Bombieri、Masser 和 Zannier 的有界高度猜想指出，对于半阿贝尔 $\overline{\mathbb{Q}}$-variety $G$ 的任何足够泛型的代数子变体，点的 Weil 高度都有一个上限包含在它与所有代数子群的交集（至多）在 $G$ 中具有互补维度的交集中。Habegger 已经在 $G$ 是乘法环面或阿贝尔变体的情况下证明了这一猜想。然而，如果 $G$ 是一个一般的半亚贝尔变体，他的方法就会遇到新的障碍。特别是，庞加莱可约性的缺乏意味着给定的半阿贝尔变体的商很难描述。为了克服这个问题，我们直接研究了 $G$ 上的某些线丛族。这使我们能够证明一般半阿贝尔变体的猜想。