There is an exercise (1.1.16) in Huybrechts: the polidisc $B_{(1,1)}(0)\subset\mathbb C^2$ and the unit disc $D$ in $\mathbb C^2$ can not be biholomorphic.
The hint is to compare the automorphisms of these two sets. The unitary matrices is a subgroup of the group of biholomorphic maps of $D$ which leaves the origin fixed. Clearly, the group of unitary matrices of $\dim 2$ is not abelian and the group of biholomorphic maps of $B_{(1,1)}(0)$ is transitive. If we can show the group of biholomorphic maps of $B_{(1,1)}(0)$ which leave invariant the origin is abelian, then this will complete the proof.
If we set $f_1(z_1,z_2)=(z_2,z_1)$ and $f_2(z_1,z_2)=(e^{i\pi/3}z_1,e^{i2\pi/3}z_2)$, then these are clearly in biholomorphic maps of $B_{(1,1)}(0)$. However, $f_1\circ f_2\neq f_2\circ f_1$.
Please tell me where is the problem. Any hint is helpful. Thx.