On the Birational Nature of Lifting

Let $X$ and $Y$ be proper birational varieties, say with only rational double points over a perfect field $k$ of positive characteristic. If $X$ lifts to $W_n(k)$, is it true that $Y$ has the same lifting property? This is true for smooth surfaces, but we show by example that this is false for smooth varieties in higher dimension, and for surfaces with canonical singularities. We also answer a stacky analogue of this question: given a canonical surface $X$ with minimal resolution $Y$ and stacky resolution $\mathcal{X}$, we characterize when liftability of $Y$ is equivalent to that of $\mathcal{X}$. The main input for our results is a study of how the deformation functor of a canonical surface singularity compares with the deformation functor of its minimal resolution. This extends work of Burns and Wahl to positive characteristic. As a byproduct, we show that Tjurina's vanishing result fails for every canonical surface singularity in every positive characteristic.