Order 3 symplectic automorphisms on K3 surfaces

Abstract

The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice \(\Lambda _{K3}\), isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps \(\pi _*\) and \(\pi ^*\) induced in cohomology by the rational quotient map \(\pi :X\dashrightarrow Y\), where X is a K3 surface admitting an order 3 symplectic automorphism \(\sigma \) and Y is the minimal resolution of the quotient \(X/\langle \sigma \rangle \); we deduce the relation between the Néron–Severi group of X and the one of Y. Applying these results we describe explicit geometric examples and generalize the Shioda–Inose structures.