Degeneration of K3 surfaces with non-symplectic automorphisms

TL;DR

This paper proves that certain K3 surfaces with specific non-symplectic automorphisms do not degenerate, relying on the rationality of automorphism actions on cohomology and assuming Kulikov models.

Contribution

It establishes non-degeneration results for K3 surfaces with primitive automorphisms of specific orders, extending understanding of their degeneration behavior.

Findings

K3 surfaces with automorphisms of order m ≠ 1,2,3,4,6 do not degenerate.

Automorphism actions on cohomology are rational.

The proof relies on the existence of Kulikov models.

Abstract

We prove that a K3 surface with an automorphism acting on the global $2$-forms by a primitive $m$-th root of unity, $m \neq 1,2,3,4,6$, does not degenerate (assuming the existence of the so-called Kulikov models). A key result used to prove this is the rationality of the actions of automorphisms on the graded quotients of the weight filtration of the $l$-adic cohomology groups of the surface.