The perfect cone compactification of quotients of type IV domains

Abstract. The perfect cone compactification is a toroidal compactification which can be defined for locally symmetric varieties. Let D_L/Õ+(L)^p be the perfect cone compactification of the quotient of the type IV domain D_L associated to an even lattice L. In our main theorem we show that the pair (D_L/Õ+(L)^p , ∆/2) has klt singularities, where ∆ is the closure of the branch divisor of D_L/Õ+(L)^p. <br>In particular this applies to the perfect cone compactification of the moduli space of 2d-polarised K3 surfaces with ADE singularities when d is square-free.