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 $X$?
