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

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 $\ge 2$. 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.