Automorphism groups of $P^1$-bundles over a non-uniruled base

TL;DR

This survey explores the automorphism groups of holomorphic $P^1$-bundles over non-uniruled compact K"ahler manifolds, with a focus on complex tori, analyzing their Jordan properties and automorphism structures.

Contribution

It provides a comprehensive discussion on the Jordan properties of automorphism groups of $P^1$-bundles over non-uniruled manifolds, especially complex tori, highlighting new insights into their structure.

Findings

Automorphism groups exhibit Jordan properties in this context.

Special behaviors are identified when the base is a complex torus.

The structure of automorphism and bimeromorphic groups is characterized.

Abstract

In this survey we discuss holomorphic $\mathbb{P}^1$-bundles $p: X \to Y$ over a non-uniruled complex compact K\"ahler manifold $Y$, paying a special attention to the case when $Y$ is a complex torus. We discuss so called Jordan properties of the groups $Aut(X)$ and $Bim(X)$ of its biholomorphic and bimeromorphic automorphisms, respectively.