Consider the quotient space of $\mathbf{CP}^2$ obtained by collapsing a line (a $\mathbf{CP}^1$) to a point. Is this a complex analytic space (in a natural way)? 



Is $\mathbb{CP}^2$ with a line collapsed a complex analytic space?
The answer is *no*, because of the following general result.



> 
> **Theorem (Grauert's contractibility criterion).** Let $X$ be a smooth complex surface and let $E \subset X$ be a connected curve in $X$, with irreducible components $E\_i$. Then there exists an analytic contraction $$\pi \colon X \to Y$$
>  of $E$ to a point $p \in Y$, where $Y$ is a (possibly singular) complex surface, if and only if the intersection matrix $(E\_i \cdot E\_j)$ is negative defined.
> 
> 
> 


If $E \subset \mathbb{CP}^2$ is a line, then $E^2=1$ and so $E$ cannot be analytically contracted to a point.


For a reference, see [K. Matsuki, *Introduction to the Mori program*, Theorem 4-6-25 p.234]. 


