Symmetric symplectic homotopy K3 surfaces

TL;DR

This paper explores how symmetries influence the smooth structures of symplectic homotopy K3 surfaces, introducing a measure of exoticness and showing that certain symmetries constrain the surface to be minimally exotic.

Contribution

It introduces a measure of exoticness for symplectic homotopy K3 surfaces and analyzes how symplectic symmetries restrict their smooth structures, revealing that maximal symmetry implies minimal exoticness.

Findings

Maximal symplectic K3 group actions force minimal exoticness.

Standard K3 is the only known minimally exotic example.

Finite symplectic symmetry groups have constrained structures.

Abstract

A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is shown that an effective action by various maximal symplectic K3 groups forces the corresponding homotopy K3 surface to be minimally exotic with respect to our measure. (However, the standard K3 is the only known example of such minimally exotic homotopy K3 surfaces.) The possible structure of a finite group of symplectic symmetries of a minimally exotic homotopy K3 surface is determined and future research directions are indicated.