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 ${\overline{D_L/{\tilde{O}}^{+}(L)}}^p$ 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 $({\overline{D_L/{\tilde{O}}^{+}(L)}}^p,\Delta /2)$ has klt singularities, where $\Delta$ is the closure of the branch divisor of $D_L/{\tilde{O}}^{+}(L)$. 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.

arXiv commentLemma 4.3 corrected. This affects the main result. To appear in Manuscripta Mathematica