Non-projective K3 surfaces with real or Salem multiplication

2025/11/25 by Bayer-Fluckiger, Eva, van Geemen, Bert, Schütt, Matthias
Mathematics · #11E12 #11R06 #14C30 #14D07 #14J28 #14J42 #14J50 #32G20 #37B40 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2511.19970

openalex publication_date 2025/11/25 · openalex created_date 2025/11/28 · openalex updated_date 2026/07/28

Abstract

We determine the Hodge endomorphism algebras of non-projective complex K3 surfaces (and more generally, hyperkähler 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.

Citations

• K3 surfaces with real or complex multiplication
• Lattice isometries and K3 surface automorphisms: Salem numbers of degree 20
• Automorphisms of K3 surfaces, signatures, and isometries of lattices
• Isometries of lattices and Hasse principles
• Automorphisms of even unimodular lattices and equivariant Witt groups
• The third smallest Salem number in automorphisms of K3 surfaces
• Real multiplication on K3 surfaces and Kuga Satake varieties
• On the Beauville form of the known irreducible symplectic varieties
• Bimeromorphic automorphism groups of non-projective hyperkähler manifolds - a note inspired by C. T. McMullen

Related

• Real multiplication on K3 surfaces and Kuga Satake varieties
• K3 surfaces with real or complex multiplication
• The Hodge conjecture for self-products of certain K3 surfaces
• Explicit families of K3 surfaces having real multiplication
• K3 surfaces of finite height over finite fields