Compactifications of moduli spaces of K3 surfaces with a higher-order nonsymplectic automorphism

TL;DR

This paper studies various compactifications of moduli spaces of K3 surfaces with specific nonsymplectic automorphisms, revealing their structure in terms of root lattices and providing explicit descriptions.

Contribution

It provides a detailed description of Baily-Borel, toroidal, and KSBA compactifications for K3 surfaces with automorphisms of orders 3 and 4, especially when fixed loci contain higher-genus curves.

Findings

Toroidal and KSBA compactifications are described via ADE root lattices.

All maximal-dimensional families for order 3 automorphisms are treated.

Explicit geometric and lattice-theoretic descriptions of the compactifications are provided.

Abstract

We describe Baily-Borel, toroidal, and geometric -- using the KSBA stable pairs -- compactifications of some moduli spaces of K3 surfaces with a nonsymplectic automorphism of order $3$ and $4$ for which the fixed locus of the automorphism contains a curve of genus $\ge2$. For order $3$, we treat all the maximal-dimensional such families. We show that the toroidal and the KSBA compactifications in these cases admit simple descriptions in terms of certain $ADE$ root lattices.