Why is this construction a complex manifold? I am beginning to study blow-ups, and in the development of the blow-up of $\mathbb C ^2$ in the origin, the author claims without further clarification that the following construction yields a complex surface: 

Consider two copies of $\mathbb C ^2,$ denoted by $U_1$ and $U_2,$ with coordinates $(x,t)$ and $(u,y),$ respectively. With the biholomorphism $$ f: U_1\backslash\{t=0\} \to U_2 \backslash\{u = 0\} $$
$$(x,t) \mapsto (1/t,tx) $$
  we identify the points of $ U_1\backslash\{t=0\} $ and $U_2 \backslash\{u = 0\} $ and construct a complex surface $\tilde C^2$

Now, I'm not sure how this is a complex manifold. It seems there is some adjunction space construction $U_1 \cup_f U_2$ involved, but how can it be given a topology and charts which will turn the set into a 2 dimensional complex manifold?
Any help is appreciated, thanks.
 A: I may be two years late, but here is an answer. As pointed out in the comment section, this really is a construction by charts.
First, let us recall how the $2$ dimensionnal sphere $\mathbb{S}^2$ can be endowed with a complex structure. The two stereographic projections (from the north and south poles) give an atlas of two charts, namely $\left\{\left(\mathbb{S}^2\setminus\{N\}, \phi_N \right),\left(\mathbb{S}^2\setminus\{S\},\phi_S \right) \right\}$, with
\begin{align}
\phi_N : \mathbb{S}^2\setminus \{N\} & \overset{\sim}\to \mathbb{R}^2, & \phi_S : \mathbb{S}^2\setminus \{S\} & \overset{\sim}\to \mathbb{R}^2,
\end{align}
and, the transition function is a function:
$$
\phi_{N,S} = \phi_N \circ {\phi_{S}}^{-1} : \mathbb{R}^2\setminus\{0\} \overset{\sim}\to \mathbb{R}^2\setminus\{0\}.
$$
(this is because $\phi_N(S)=0$ and $\phi_S(N)= 0$.) What is miraculous here is that, identifying $\mathbb{R}^2$ with $\mathbb{C}$, this transition function just becomes:
\begin{align}
\mathbb{C}^* & \longrightarrow \mathbb{C}^* \\
z & \longmapsto \frac{1}{z}
\end{align}
and hence, is holomorphic. This provides an holomorphic atlas for $\mathbb{S}^2$, and hence, it is a complex manifold. More: this complex manifold is uniquely described by the fact it is the union of two copies of $\mathbb{C}$, say $U_1$ and $U_2$, glued together along the map $z \in \mathbb{C}^* \mapsto \frac 1 z \in \mathbb{C}^*$.
The same thing appears here. The manifold $\tilde{C}^2$ is uniquely described as the union of two copies of $\mathbb{C}^2$ glued together along the biholomorphism $f$ you described. Rigorously, $\tilde{C}^2$ is given by an atlas of two charts, say $\left\{ \left(U_1,f_1\right),\left(U_2,f_2\right)\right\}$, with $f_i : U_i \overset{\sim}\to \mathbb{C}^2$ and the transition function $f_1\circ f_2^{-1}= f : \mathbb{C}^2 \setminus  \mathbb{C}\times\{0\} \overset{\sim}\to \mathbb{C}^2\setminus\{0\}\times \mathbb{C}$ defined by $f(x,t) = (1/t,tx)$, which is holomorphic.
