The perfect cone compactification of quotients of type IV domains

DOI10.1007/S00229-021-01356-WarXiv1904.08638MaRDI QIDQ6317406

Luca Giovenzana

Publication date: 18 April 2019

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.

Mathematics Subject Classification ID

(K3) surfaces and Enriques surfaces (14J28) Families, moduli, classification: algebraic theory (14J10)

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