Abstract
The perfect cone compactification is a toroidal compactification which can be defined for locally symmetric varieties. Let overline{D_{L}/widetilde{O}^{+}(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 { (overline{D_{L}/widetilde{O}^{+}(L)}^{p}, Delta /2) } has klt singularities, where Delta is the closure of the branch divisor of { D_{L}/widetilde{O}^{+}(L) }. 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.
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