  The moduli space for polarized hyperk\"ahler manifolds of
$\mathrm{K3}^{[m]}$-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.
