When the geometric invariant theory and the Baily-Borel theory both apply to a moduli space, the two resulting compactifications do not necessarily coincide. They usually differ by a birational transformation in terms of an arrangement, for which the general pattern was firstly described by Looijenga. In this paper, we compare the MMP singularities on both sides for the ball quotient case. We show that if the relevant arrangement is nonempty, the birational transformation from the GIT compactification to the Baily-Borel compactification turns non-log canonical singularities to log canonical singularities. We illustrate this with the moduli spaces of quartic curves, rational elliptic surfaces and cubic threefolds.
Dali Shen (Mon,) studied this question.