Symmetries and exotic smooth structures on a $K3$ surface

TL;DR

This paper investigates the symmetries of exotic and standard K3 surfaces, revealing restrictions on group actions, proving nonsmoothability of certain automorphisms, and explicitly describing fixed-point structures of symplectic actions.

Contribution

It establishes strong restrictions on finite group actions on exotic K3 surfaces, proves nonsmoothability of certain automorphisms, and characterizes fixed-point sets of symplectic cyclic actions.

Findings

Finite group actions on exotic K3 surfaces are highly restricted.

Automorphisms of prime order ≥7 cannot be realized smoothly.

Fixed-point set structures for symplectic cyclic actions of prime order ≥5 are explicitly determined.

Abstract

Smooth and symplectic symmetries of an infinite family of distinct exotic $K3$ surfaces are studied, and comparison with the corresponding symmetries of the standard $K3$ is made. The action on the $K3$ lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability of actions induced by a holomorphic automorphism of a prime order $\geq 7$ is proved and nonexistence of smooth actions by several $K3$ groups is established (included among which is the binary tetrahedral group $T_{24}$ which has the smallest order). Concerning symplectic symmetries, the fixed-point set structure of a symplectic cyclic action of a prime order $\geq 5$ is explicitly determined, provided that the action is homologically nontrivial.