For a path-connected differentiable manifold $M$, the group of diffeomorphisms $\text{Diff}(M)$ acts on $M$ transitively. In fact for each pair of points $x,y \in M$, there is a one-parameter family $\Phi:M \times [0,1] \rightarrow M$ such that $\Phi$ is continuous, $\forall t, \Phi_t \in \text{Diff}(M)$, and $\Phi_0(x)=0, \Phi_1(x)=y$. This family can be constructed by moving points on manifold locally.
Such transportation can firstly be performed on $\mathbb{R}^n$ while fixing every point outside a ball by creating a constant flow weighted by a bump function and solving an ODE for this flow. Then for any path connecting $x$ and $y$, we can cover the path by finitely many Euclidean balls and perform the local transportation on each ball.
The natural question is to ask what happens for a complex manifold $X$. For any pair of points on a path-connected complex manifold, does the group of biholomorphisms on $X$ act transitively on $X$? If not, what is the orbit of each point? The same argument above will not work since we can't work with bump functions on a complex manifold.