K3 surfaces with two involutions and low Picard number

Abstract

Let $X$ be a complex algebraic K3 surface of degree $2d$ and with Picard number $\rho$. Assume that $X$ admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, $\rho \geq 1$ when $d=1$ and $\rho \geq 2$ when $d \geq 2$. For $d=1$, the first example defined over $\mathbb{Q}$ with $\rho=1$ was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kond\={o}, also defined over $\mathbb{Q}$, can be used to realise the minimum $\rho=2$ for all $d\geq 2$. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum $\rho=2$ for $d=2,3,4$. We also show that a nodal quartic surface can be used to realise the minimum $\rho=2$ for infinitely many different values of $d$. Finally, we strengthen a result of Morrison by showing that for any even lattice $N$ of rank $1\leq r \leq 10$ and signature $(1,r-1)$ there exists a K3 surface $Y$ defined over $\mathbb{R}$ such that $\textrm{Pic} Y_\mathbb{C}=\textrm{Pic} Y \cong N$.