TITLE: Is $\mathbb{CP}^2$ with a line collapsed a complex analytic space?
QUESTION [6 upvotes]: 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)?

REPLY [10 votes]: 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].