Partial Hasse invariants, partial degrees and the canonical subgroup

If the Hasse invariant of a $p$-divisible group is small enough, then one can construct a canonical subgroup inside its $p$-torsion. We remark that, assuming the existence of a subgroup of adequate height in the $p$-torsion whose dual has small degree, the expected properties of the canonical subgroup can be easily proven. A fundamental relation is the equality between the Hasse invariant and the degree of the dual of the canonical subgroup. When one considers a $p$-divisible group $G$ with an action of the ring of integers of a (possibly ramified) finite extension of $\mathbb{Q}_p$, then much more can be said. One can define partial Hasse invariants ; they are natural in the unramified case, and generalize a construction of Reduzzi and Xiao in the general case. One can also define partial degrees for finite flat subgroups of $G$. We prove some properties for these partial Hasse invariants and partial degrees, and compute the partial degrees of the canonical subgroup.