  We prove two finiteness results for reductions of Hecke orbits of abelian
varieties over local fields: one in the case of supersingular reduction and one
in the case of reductive monodromy. As an application, we show that only
finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which
in particular implies that in each fixed dimension $g$ only finitely many
supersingular abelian varieties admit CM lifts. Combining this with the
Kuga-Satake construction, we also show that only finitely many supersingular
$K3$-surfaces admit CM lifts. Our tools include $p$-adic Hodge theory and group
theoretic techniques.
