K3 surfaces with real or complex multiplication

TL;DR

This paper constructs families of complex projective K3 surfaces with specified real or complex multiplication properties, extending to hyperkähler manifolds, under certain degree constraints.

Contribution

It demonstrates the existence of families of K3 surfaces with real or complex multiplication for degrees satisfying md ≤ 21, broadening understanding of their moduli.

Findings

Existence of (m-2)-dimensional families with real multiplication by E.

Analogous constructions for CM fields.

Extension of results to hyperkähler manifolds.

Abstract

Let $E$ be a totally real number field of degree $d$ and let $m \geqslant 3$ be an integer. We show that if $md \leqslant 21$ then there exists an $(m-2)$-dimensional family of complex projective $K3$ surfaces with real multiplication by $E$. Analogous results are proved for CM number fields and also for all known higher-dimensional hyperk\"ahler manifolds.