$SL(2)/U(2)$ is isomorphic to $\mathbb{C}^2 \setminus \left\{ 0 \right\}$? Let  $G= SL(2)$ and $H=U(2)$ the set of $2x2$ matrices of type $\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix}$. By Chevalley Theorem $G/H$ can be immersed in $\mathbb{P}(\mathbb{C}^2 \oplus \mathbb{C})$ and can be described as the set of $g \cdot (\mathbb{C}e_1 \oplus \mathbb{C})$ with $g \in G$. Why is $G/H$ isomorphic to $\mathbb{C}^2 \setminus \left\{ 0 \right\}$? They are both quasiprojective varieties, so I should find an open covering of affine sets $U_i \subset SL(2)$ such that if $\phi : G/H \to \mathbb{C}^2 \setminus \left\{ 0 \right\}$, $\phi|_{U_i}$ is an isomorphism of affine varieties (therefore regular and invertible). Any hints?
 A: Since Sasha probably won't be more specific I'll add some more details to the answer. Although the content is virtually the same the wording might be a bit more psychologically comforting.
There is a natural action of $G = \mathrm{GL}_2(\mathbb{C})$ on $\mathbb{C}^2 \setminus \{0\}$.
Exercise. This action of $G$ on $\mathbb{C}^2 \setminus \{0\}$ is transitive, meaning that, if you pick two points $x,y \in \mathbb{C}^2 \setminus \{0\}$, you can always find a $g \in G$ such that $gx = y$.
Now consider the point $x = (1, 0)$. By the orbit--stabilizer theorem, there should be an isomorphism between $G / G_x$ (the quotient of $G$ by the stabilizer at $x$) and $Gx$ (the orbit of $x$).
Exercise. In case you've only seen the orbit--stabilizer theorem for discrete groups, prove that it also works in the expected way for Lie groups acting on topological spaces.
Exercise. Prove that the stabilizer of the point $x = (1, 0)$ is isomorphic to $H$. Also prove that the orbit $Gx$ is all of $\mathbb{C}^2 \setminus \{0\}$. Conclude the desired result.
