This question is similar to, but not the same as this one. Take the space of automorphisms of $\mathbb{C}^n$ in the holomorphic category, with the compact-open topology. For $n=1$ this is just $\mathbb{C} \times \mathbb{C}^*$, but for larger $n$ is very complicated. It clearly contains $GL(n,\mathbb{C})$, and translations.

To give a taste of how big $Aut(\mathbb{C}^n)$ is, there is a theorem that given any two countable dense subsets $X,Y \subset \mathbb{C}^n$, $n >1$, there is a volume preserving automorphism taking $X$ to $Y$. But I have no idea about what this space is like. Is it some sort of infinite dimensional manifold? Analytic space? Does it contain an infinite dimensional Lie group as a (closed) subgroup?

finite typeconnected smooth group variety. This forces the points of $G^0$ to act by automorphisms represented by polynomials ofboundeddegree, which is absurd for $n > 1$ since for two distinct variables $(x,y)$ can use $(x,y) \mapsto (x + t y^n, y)$ to get a connected family of auts joining id to arb. big degree. $\endgroup$ – BCnrd Dec 1 '10 at 5:06