Suppose we are given a smooth complex algebraic variety $X$ with a smooth, irreducible divisor $D$ such that $D$ is Fano and the normal bundle $L$ to $D$ is anti-ample. Then we can contract $D$ to a point, at least in the category of complex analytic spaces.

Is the analytic isomorphism type of the singularity determined by $L$, or does it depend on higher order infinitesimal information of $D$ inside $X$?

onlydepends on the normal bundle in my question; for $\mathbb{P}^1$, you're saying it can depend on higher order information as well? – MStiz Sep 5 '12 at 14:03