<jats:p>The perfect cone compactification is a toroidal compactification which can be defined for locally symmetric varieties. Let <jats:inline-formula><jats:alternatives><jats:tex-math>$$\overline{D_{L}/\widetilde{O}^{+}(L)}^{p}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mover>
<mml:mrow>
<mml:msub>
<mml:mi>D</mml:mi>
<mml:mi>L</mml:mi>
</mml:msub>
<mml:mo>/</mml:mo>
<mml:msup>
<mml:mover>
<mml:mi>O</mml:mi>
<mml:mo>~</mml:mo>
</mml:mover>
<mml:mo>+</mml:mo>
</mml:msup>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>L</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mo>¯</mml:mo>
</mml:mover>
<mml:mi>p</mml:mi>
</mml:msup>
</mml:math></jats:alternatives></jats:inline-formula> be the perfect cone compactification of the quotient of the type IV domain <jats:inline-formula><jats:alternatives><jats:tex-math>$$D_{L}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>D</mml:mi>
<mml:mi>L</mml:mi>
</mml:msub>
</mml:math></jats:alternatives></jats:inline-formula> associated to an even lattice <jats:italic>L</jats:italic>. In our main theorem we show that the pair <jats:inline-formula><jats:alternatives><jats:tex-math>$${ (\overline{D_{L}/\widetilde{O}^{+}(L)}^{p}, \Delta /2) }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:msup>
<mml:mover>
<mml:mrow>
<mml:msub>
<mml:mi>D</mml:mi>
<mml:mi>L</mml:mi>
</mml:msub>
<mml:mo>/</mml:mo>
<mml:msup>
<mml:mover>
<mml:mi>O</mml:mi>
<mml:mo>~</mml:mo>
</mml:mover>
<mml:mo>+</mml:mo>
</mml:msup>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>L</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mo>¯</mml:mo>
</mml:mover>
<mml:mi>p</mml:mi>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:mi>Δ</mml:mi>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> has klt singularities, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Delta $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>Δ</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> is the closure of the branch divisor of <jats:inline-formula><jats:alternatives><jats:tex-math>$${ D_{L}/\widetilde{O}^{+}(L) }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mi>D</mml:mi>
<mml:mi>L</mml:mi>
</mml:msub>
<mml:mo>/</mml:mo>
<mml:msup>
<mml:mover>
<mml:mi>O</mml:mi>
<mml:mo>~</mml:mo>
</mml:mover>
<mml:mo>+</mml:mo>
</mml:msup>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>L</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula>. In particular this applies to the perfect cone compactification of the moduli space of 2<jats:italic>d</jats:italic>-polarised <jats:italic>K</jats:italic>3 surfaces with ADE singularities when <jats:italic>d</jats:italic> is square-free.</jats:p>
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