Let $X$ be a proper smooth variety over a $p$-adic field $K$. Let $\mathcal{O}_K$ be the ring of integers of $K$ and $k$, its residue field. We say that $X$ has good ordinary reduction if there is a smooth proper model $\mathfrak{X}$ of $X$ over $\mathcal{O}_K$ such with special fibre $Y$ satisfying $H^r (Y, d\Omega_Y^s) = 0$ for all $r$ and $s$, where $d\Omega_Y^s$ is the sheaf of exact differentials on $Y$. This definition is due to Kato, following his paper with S. Bloch [$p$-adic etale cohomologies]. When $X$ is a $d$-dimensional abelian variety, the above definition coincides with the classical definition of ''ordinary" for abelian varieties: that the group of $\bar{k}$-points killed by $p$ on $Y$ has order $p^d$.

Now, for an elliptic curve $E$, we say it has good supersingular reduction if the group of $\bar{k}$-points of the reduction of $E$ killed by $p$ is trivial. (I don't know if this definition is the same for abelian varieties; I would be grateful if somebody will confirm or correct me on this).

**Question**: Is there a general notion of good supersingular reduction for a proper smooth variety $X$ over a $p$-adic field $K$ as above?