K3 surfaces with a symplectic automorphism of order 4

Given X$X$, a K3 surface admitting a symplectic automorphism τ$\tau$ of order 4, we describe the isometry τ∗$\tau ^*$ on H2(X,Z)$H^2(X,\mathbb {Z})$. Having called Z∼$\tilde{Z}$ and Y∼$\tilde{Y}$, respectively, the minimal resolutions of the quotient surfaces Z=X/τ2$Z=X/\tau ^2$ and Y=X/τ$Y=X/\tau$, we also describe the maps induced in cohomology by the rational quotient maps X→Z∼,X→Y∼$X\rightarrow \tilde{Z},\ X\rightarrow \tilde{Y}$ and Y∼→Z∼$\tilde{Y}\rightarrow \tilde{Z}$: With this knowledge, we are able to give a lattice‐theoretic characterization of Z∼$\tilde{Z}$, and find the relation between the Néron–Severi lattices of X,Z∼$X,\tilde{Z}$ and Y∼$\tilde{Y}$ in the projective case. We also produce three different projective models for X,Z∼$X,\tilde{Z}$ and Y∼$\tilde{Y}$, each associated to a different polarization of degree 4 on X$X$.