On the image of the period map for polarized hyperk\"ahler manifolds

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.