# Pink's conjecture on unlikely intersections and families of semi-abelian varieties

## TL;DR

This paper investigates Pink's conjecture on unlikely intersections within mixed Shimura varieties, providing counterexamples to the relative Manin-Mumford conjecture and supporting Pink's broader conjecture.

## Contribution

It demonstrates that special subvarieties of mixed Shimura varieties cannot all be characterized by group subschemes, offering new insights into unlikely intersection conjectures.

## Key findings

- Counter-example to the relative Manin-Mumford conjecture.
- Evidence supporting Pink's conjecture on unlikely intersections.
- Analysis of mixed Hodge structures and uniformization of the Poincaré torsor.

## Abstract

The Poincar\'e torsor of a Shimura family of abelian varieties can be viewed both as a family of semi-abelian varieties and as a mixed Shimura variety. We show that the special subvarieties of the latter cannot all be described in terms of the group subschemes of the former. This provides a counter-example to the relative Manin-Mumford conjecture, but also some evidence in favour of Pink's conjecture on unlikely intersections in mixed Shimura varieties. The main part of the article concerns mixed Hodge structures and the uniformization of the Poincar\'e torsor, but other, more geometric, approaches are also discussed.