Let $ Y \subset X$ be a closed subspace of an algebraic space of finite type over $\mathbb{C}$. Let $p : Y' \rightarrow Y$ be a proper map of algebraic spaces. Artin proved that there exists a birational modificatiom (dilatation) $q : X' \rightarrow X$ which induces $p$ when restricted to $Y'$ if and only if this dilatation exists over the formal completion $\mathcal{X}$ of $X$ along $Y$.
In case we are not dealing with dilatation, but rather with contraction (that is a map $Y \rightarrow Y'$), Artin gives several cohomological criteria on the map $Y \rightarrow Y'$ for the formal contraction $\mathcal{X} \rightarrow \mathcal{X}'$ to exist.
But it seems he does not discus too much the case of a dilatations. Are there some (cohomological, geometric, topological) criteria on the spaces $Y$, $Y'$ and the map $p : Y' \rightarrow Y $ for a formal dilatation $\mathcal{X}' \rightarrow \mathcal{X}$ (which induces $p$ wheb restricted to $Y'$) to exist?
Many thanks in advance.
