# Complex tori, theta groups and their Jordan properties

## TL;DR

This paper investigates the limitations of Jordan's theorem in the context of bimeromorphic automorphism groups of certain complex manifolds, specifically products of projective lines and complex tori.

## Contribution

It demonstrates that an analogue of Jordan's theorem fails for these automorphism groups, highlighting differences from classical linear group behavior.

## Key findings

- Jordan's theorem does not extend to automorphism groups of certain complex manifolds.
- The group of bimeromorphic automorphisms of a product of the projective line and a complex torus is not Jordan.
- This reveals new structural properties of automorphism groups in complex geometry.

## Abstract

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does not hold for the group of bimeromorphic automorphisms of a product of the complex projective line and a complex torus of positive algebraic dimension.