Let $\mathbb P_1$ be the one dimensional complex projective space.
What is the connected component of the full automorphism of 
$\mathbb C^*\times \mathbb P_1$.
Is it a complex Lie group? I mean is it finite dimensional?

We know that ${\rm Aut}(\mathbb C)^\ast$ is ${\mathbb C}^\ast$, and $Aut^\circ(\mathbb P_1)$ is $PSL(2,\mathbb C)$.