Fixed-locus relations and purely non-symplectic automorphisms of order 6 on K3 surfaces

Abstract

We study relations between invariants of the fixed loci of purely non-symplectic automorphisms of $K3$ surfaces. We obtain these relations by comparing the stringy Euler characteristic of higher dimensional Borcea--Voisin varieties with their Hodge numbers. For orders $4$ and $6$ we explain the geometric meaning of these relations and show how they follow from the Lefschetz and Riemann--Hurwitz formulas. For order $6$, the classifications of fixed loci for orders $2$ and $3$ give $817$ candidates. We use local conditions on discriminant forms, Hermitian trace forms and bounds for root-free lattices to reduce this number to $204$. All $142$ numerical types obtained from the $150$ deformation classes of Brandhorst and Hofmann satisfy these conditions. Also, we give explicit elliptic and projective models for each of the $20$ possible fixed loci of the generator.

17 pages. Substantially revised and expanded; title changed. Fixed-locus relations corrected and clarified. New explicit models and a lattice sieve for order 6. Code and tables included as ancillary files. arXiv admin note: substantial text overlap with arXiv:2107.04104