K3 surfaces with a symplectic automorphism of order 4

Abstract

Given $X$ a K3 surface admitting a symplectic automorphism $\tau$ of order 4, we describe the isometry ${\tau }^{\ast }$ on $H^2(X,\mathbb{Z})$. Having called $\tilde{Z}$ and $\tilde{Y}$ respectively the minimal resolutions of the quotient surfaces $Z=X/{\tau }^2$ and $Y=X/\tau$, we also describe the maps induced in cohomology by the rational quotient maps $X\to \tilde{Z},X\to \tilde{Y}$ and $\tilde{Y}\to \tilde{Z}$: with this knowledge, we are able to give a lattice-theoretic characterization of $\tilde{Z}$, and find the relation between the Néron-Severi lattices of $X,\tilde{Z}$ and $\tilde{Y}$ in the projective case. We also produce three different projective models for $X,\tilde{Z}$ and $\tilde{Y}$, each associated to a different polarization of degree 4 on $X$.

38 pages