# The perfect cone compactification of quotients of type IV domains

## TL;DR

This paper proves that the perfect cone compactification of certain quotients of type IV domains has mild singularities, specifically klt singularities, with applications to moduli spaces of polarized K3 surfaces.

## Contribution

It establishes that the pair of the perfect cone compactification and the branch divisor has klt singularities, extending to moduli spaces of polarized K3 surfaces with ADE singularities.

## Key findings

- The pair has klt singularities.
- Applies to moduli spaces of 2d-polarized K3 surfaces with square-free d.
- Provides a new understanding of the singularities in toroidal compactifications.

## 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.