Complex Tori and Automorphism Groups of certain $P^1$-bundles

This paper is a continuation of our article (European J. Math., https://doi.org/10.1007/s40879-020-00419-8). The notion of a poor complex compact manifold was introduced there and the group $Aut(X)$ for a $P^1$-bundle over such a manifold was proven to be very Jordan. We call a group $G$ very Jordan if it contains a normal abelian subgroup $G_0$ such that the orders of finite subgroups of the quotient $G/G_0$ are bounded by a constant depending on $G$ only. In this paper we provide explicit examples of infinite families of poor manifolds of any complex dimension, namely simple tori of algebraic dimension zero. Then we consider a non-trivial holomorphic $P^1$-bundle $(X,p,Y)$ over a non-uniruled complex compact Kaehler manifold $Y$. We prove that $Aut(X)$ is very Jordan provided some additional conditions on the set of sections of $p$ are met. Applications to $P^1$-bundles over non-algebraic complex tori are given.