$SL(2)/U(2)$ 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?

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.
