  In the moduli space of polarized varieties the same unpolarized variety can
occur multiple times However, for K3 surfaces, compact hyperk\"ahler manifolds,
and abelian varieties the number is finite. This may be viewed as a consequence
of the Kawamata-Morrison cone conjecture. In this note we provide a proof of
this finiteness not relying on the cone conjecture and, in fact, not even on
the global Torelli theorem. Instead, it uses the geometry of the moduli space
of polarized varieties to conclude the finiteness by means of Baily-Borel type
arguments. We also address related questions concerning finiteness in twistor
families associated with polarized K3 surfaces of CM type.
