The perfect cone compactification of some quotients of type IV domains has canonical singularities

The quotient of a type IV hermitian symmetric domain $D$ by the action of an arithmetic group of automorphisms $\Gamma$ is a quasi projective variety by a result of Baily and Borel. Compactifications of it have been widely studied. While the Baily-Borel compactification has a highly singular boundary, it has been proven that smooth toroidal compactifications exist. In this talk I will introduce a specific toroidal compactification, namely the perfect cone compactification and explain why, when the group $\Gamma$ is the stable orthogonal group acting on $D$, it has canonical singularities. As a corollary we get that the perfect cone compactification of the moduli space of semipolarised K3 surfaces has canonical singularities.