# The Bounded Height Conjecture for Semiabelian Varieties

## TL;DR

This paper proves the Bounded Height Conjecture for general semiabelian varieties by analyzing line bundles directly, overcoming previous obstructions related to quotients and Poincaré reducibility.

## Contribution

It introduces a new approach using families of line bundles to establish the conjecture for all semiabelian varieties, extending prior results.

## Key findings

- Proves the Bounded Height Conjecture for all semiabelian varieties.
- Develops a method based on line bundles to handle complex quotients.
- Overcomes previous obstructions related to Poincaré reducibility.

## Abstract

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.