  The moduli space for polarized hyperk\"ahler manifolds of
$\mathrm{K3}^{[m]}$-type or $\mathrm{Kum}_m$-type with a given polarization
type is not necessarily connected, which is a phenomenon that only happens for
$m$ large. The period map restricted to each connected component gives an open
embedding into the period domain, and the complement of the image is a finite
union of Heegner divisors. We give a simplified formula for the number of
connected components, as well as a simplified criterion to enumerate the
Heegner divisors in the complement. In particular, we show that the image of
the period map may be different when restricted to different components of the
moduli space.
