Suppose we have a proper (or projective) singular variety $X_0$ over a field of characteristic $p$, $k$, and with dual graph $\Gamma$. We may suppose that the components of $X_0$ are smooth and reduced, irreducible (if you wish a simple normal crossing variety).
I wonder about the lifting problem:
There exist a flat proper scheme $\mathcal X$ over the spectrum of some local domain of mixed characteristic $(o,p)$ such that the special fibre is isomorphic to $X_0$ and whose generic fibre is singular with the same dual graph of $X$?
The question is: What is known about the obstruction to have such a lift.