Non-projective K3 surfaces with real or Salem multiplication

TL;DR

This paper classifies the Hodge endomorphism algebras of non-projective complex K3 surfaces, revealing they are either totally real fields or Salem number fields, contrasting with the projective case.

Contribution

It provides a complete classification of endomorphism algebras for non-projective K3 surfaces and establishes new existence criteria linking geometry, number theory, and dynamics.

Findings

Endomorphism algebras are either totally real or Salem number fields.

Contrasts with the projective case where fields are either totally real or CM.

Develops criteria for the existence of such K3 surfaces.

Abstract

We determine the Hodge endomorphism algebras of non-projective complex K3 surfaces (and more generally, hyperk\"ahler manifolds). We show that they are either totally real fields or number fields generated by Salem numbers. This is unlike the projective case, where the endomorphism fields are either totally real or CM. We also develop precise existence criteria and explore the relations to number theory and dynamics.