K3 surfaces with real or complex multiplication
Eva Bayer-Fluckiger, Bert van Geemen, Matthias Schütt
TL;DR
The paper proves that for a totally real field $E$ of degree $d$ and integers $m\ge3$ with $md\le21$, there exists a family of complex projective K3 surfaces $X$ whose Hodge endomorphism algebra $A_X$ is isomorphic to $E$ and for which ${\rm dim}_E(T_{X,{\mathbf Q}})=m$; analogous results hold for CM fields with $md\le20$ and extend to higher-dimensional hyperkähler manifolds. The authors develop a transfer framework: given a field $E$, a (hermitian or quadratic) form $W$ over $E$ transfers to a rational quadratic form $\mathrm{T}(W)$ over ${\mathbf Q}$, with explicit dimension, determinant, and local behavior; this enables a classification of when such a $U$ can arise as a transfer. The key geometric mechanism is the surjectivity of the period map for HK manifolds and K3 surfaces, together with Torelli-type results, which realize abstract $({\rm dim},\det, {\rm disc})$-compatible transfers as concrete endomorphism algebras of transcendental Hodge structures in actual geometric objects. The work further provides a coherent framework (Condition (C)) for embedding RM/CM data into HK lattices and yields explicit manifestations of RM/CM in Picard lattices and elliptic fibrations, with extensions to the known HK deformation types. These results illuminate deep links between arithmetic of endomorphism algebras, lattice theory, and the geometry of K3 and HK manifolds, with potential implications for Hodge and Tate conjectures via RM/CM symmetries.
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 hyperkähler manifolds.